107:
607:
2681:, as a way to circumvent the favoritism inherent when one party chooses before the other. An example showed how a divorcing couple might reach a fair settlement in the distribution of jointly-owned items. The parties would take turns to be the first chooser at different points in the selection process: Ann chooses one item, then Ben does, then Ben chooses one item, then Ann does.
3205:
3572:
rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw. At the time, consecutive identical board states were required to trigger the rule; the rule was later amended to the same board position reoccurring three times at any point, as the sequence shows that the
3507:
2972:
2796:
of winning exceeded their own. They proved that, as the duelers’ hitting probability approaches zero, the firing sequence converges to the Thue–Morse sequence. In so doing, they demonstrated that the Thue–Morse order produces a fair outcome not only for sequences
2813:
Thus the mathematics supports using the Thue–Morse sequence instead of alternating turns when the goal is fairness but earlier turns differ monotonically from later turns in some meaningful quality, whether that quality varies continuously or discretely.
589:
1133:
2573:
For generalizations of the Thue–Morse sequence and the
Prouhet–Tarry–Escott problem to partitions into more than two parts, see Bolker, Offner, Richman and Zara, "The Prouhet–Tarry–Escott problem and generalized Thue–Morse sequences".
2695:, proposed the Thue–Morse sequence as a way to reduce the advantage of moving first. They suggested that “it would be interesting to quantify the intuition that the Thue–Morse order tends to produce a fair outcome.”
2672:
invoked the Thue–Morse sequence but did not identify it as such. When allocating a contested pile of items between two parties who agree on the items' relative values, Brams and Taylor suggested a method they called
2452:
4787:. Encyclopedia of Mathematics and Its Applications. Vol. 105. A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot,
1554:
3200:{\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {5t_{n-1}+3t_{n}}{n^{2}}}&=4\zeta (2)={\frac {2\pi ^{2}}{3}},\\\sum _{n\geq 1}{\frac {9t_{n-1}+7t_{n}}{n^{3}}}&=8\zeta (3),\end{aligned}}}
3334:
2130:
2550:
be a multiple of 2 is not strictly necessary: there are some further cases for which a solution exists. However, it guarantees a stronger property: if the condition is satisfied, then the set of
2977:
465:
2966:
Certain linear combinations of
Dirichlet series whose coefficients are terms of the Thue–Morse sequence give rise to identities involving the Riemann Zeta function (Tóth, 2022 ). For instance:
1328:
2229:
4564:. Encyclopedia of Mathematics and Its Applications. Vol. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press.
3254:
1288:
1414:
1255:
460:
1004:
3286:
1654:
2789:
duel, in which each of the shooters has equally poor shooting skills. Cooper and Dutle postulated that each dueler would demand a chance to fire as soon as the other's
1499:
2294:
1680:
1387:
1361:
1798:
1766:
1621:
1730:
1460:
1188:
4401:
3326:
3306:
2261:
1822:
1700:
1594:
1574:
1434:
1228:
1208:
2698:
Robert
Richman addressed this problem, but he too did not identify the Thue–Morse sequence as such at the time of publication. He presented the sequences
2785:
Joshua Cooper and Aaron Dutle showed why the Thue–Morse order provides a fair outcome for discrete events. They considered the fairest way to stage a
2586:, a curve can be generated if an automaton is programmed with a sequence. When Thue–Morse sequence members are used in order to select program states:
4883:
3640:
2817:
Sports competitions form an important class of equitable sequencing problems, because strict alternation often gives an unfair advantage to one team.
2853:
2828:
4808:
4758:
4697:
4569:
4491:
4362:
4305:
Bolker, Ethan; Offner, Carl; Richman, Robert; Zara, Catalin (2016). "The
Prouhet–Tarry–Escott problem and generalized Thue–Morse sequences".
4287:
4253:
4171:
2324:
3607:
31:
4955:
4098:
4674:
4480:
Developments in
Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, California, USA, June 26-29, 2006
2371:
4109:
2831:
in soccer. He did a set of field experiments with pro players and found that the team kicking first won 60% of games using ABAB (or
2934:: captain A has the first, fourth, sixth, and seventh choices, while captain B has the second, third, fifth, and eighth choices.
4478:
Krieger, Dalia (2006). "On critical exponents in fixed points of non-erasing morphisms". In Ibarra, Oscar H.; Dang, Zhe (eds.).
3959:
3502:{\displaystyle (2^{s}+1)\sum _{n\geq 1}{\frac {t_{n-1}}{n^{s}}}+(2^{s}-1)\sum _{n\geq 1}{\frac {t_{n}}{n^{s}}}=2^{s}\zeta (s).}
1504:
813:"""Return an int containing the first 2**n bits of the Thue-Morse sequence, low-order bit 1st."""
4905:
4799:, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé. Cambridge:
4592:
2026:, which is simply the Thue–Morse sequence on (1,0) instead of on (0,1). This property may be generalized to the concept of an
2054:
70:. The first few steps of this procedure yield the strings 0, 01, 0110, 01101001, 0110100110010110, and so on, which are the
4960:
4843:
4518:
4439:
4245:
790:
275:
3602:
2883:
crew members that eliminates transverse forces (and hence sideways wiggle) on a four-membered coxless racing boat, while
4374:
2923:) would be even more fair. Richman suggested that the fairest way for “captain A” and “captain B” to choose sides for a
4126:
4838:
4191:
2906:
1801:
1293:
4800:
4750:
4279:
4163:
3984:
2145:
4950:
4724:
1987:
3964:
1828:
2818:
2625:
of infinite length containing a finite area. This illustrates the fractal nature of the Thue–Morse
Sequence.
2315:
occur for few (finitely many) positions in the game, with all remaining positions having odious nim-values.
54:(an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the
4630:
3537:
2909:
have no pre-existing imbalance to correct, so they often use a “snake” draft (forward, backward, etc.; or
2750:
2559:
71:
4833:
4914:. Freeware to generate self-similar music based on the Thue–Morse Sequence and related number sequences.
4893:
sequence A000069 (Odious numbers: numbers with an odd number of 1's in their binary expansion)
3545:
2865:
2562:
can be partitioned into two sets with equal sums. This follows directly from the expansion given by the
4901:
sequence A001969 (Evil numbers: numbers with an even number of 1's in their binary expansion)
3213:
1260:
584:{\displaystyle {\begin{aligned}t_{0}&=0,\\t_{2n}&=t_{n},\\t_{2n+1}&=1-t_{n},\end{aligned}}}
4684:
Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. (eds.).
4396:
1128:{\displaystyle \prod _{i=0}^{\infty }\left(1-x^{2^{i}}\right)=\sum _{j=0}^{\infty }(-1)^{t_{j}}x^{j},}
4796:
4271:
4210:
3827:
3569:
2790:
2756: − 1. A consequence of this result is that a resource whose value is expressed as a
1999:
1392:
1233:
3518:
3582:
3557:
2761:
2688:
2684:
2046:
678:. So, the first element is 0. Then once the first 2 elements have been specified, forming a string
452:
3259:
2916:). Ian Allan argued that a “third-round reversal” (forward, backward, backward, forward, etc.; or
1626:
4653:
4600:
4545:
4527:
4466:
4448:
4418:
4332:
4314:
4226:
4200:
4047:
3549:
2880:
2757:
2336:
2027:
1883:
2786:
4510:
2764:
is most fairly allocated using a sequence that converges to Thue–Morse as the function becomes
1465:
4852:
4804:
4754:
4693:
4670:
4565:
4487:
4358:
4283:
4249:
4167:
2692:
2301:
1866:
106:
4354:
4347:
2266:
1659:
1366:
1333:
4814:
4764:
4703:
4645:
4609:
4588:
4575:
4537:
4497:
4458:
4410:
4383:
4324:
4293:
4259:
4218:
4177:
3592:
2773:
2563:
675:
432:
The resulting algorithm takes constant time to generate each sequence element, using only a
34:
This graphic demonstrates the repeating and complementary makeup of the Thue–Morse sequence.
4621:
4038:(2022). "Linear Combinations of Dirichlet Series Associated with the Thue-Morse Sequence".
1771:
1739:
1599:
17:
4818:
4768:
4707:
4689:
4617:
4579:
4501:
4483:
4297:
4263:
4181:
4155:
3587:
3523:
2902:
2716:
2583:
1951:
1942:
55:
51:
4541:
4462:
1709:
1439:
4870:
4855:
4214:
4086:. Naval Postgraduate School, Department of Mathematics, Monterey, California, USA: 3–11.
1170:
4792:
4788:
4781:
3311:
3291:
2955:
2898:
2869:
2712:
2669:
2618:
2246:
1807:
1685:
1579:
1559:
1419:
1213:
1193:
433:
167:
134:
4431:
4944:
4906:
Reducing the influence of DC offset drift in analog IPs using the Thue-Morse
Sequence
4649:
4387:
4336:
4230:
4066:
3969:
3541:
3529:
2947:
2776:
2765:
2742:
2708:
2661:
2622:
2348:
215:
This method leads to a fast method for computing the Thue–Morse sequence: start with
130:
87:
63:
4657:
4549:
4422:
4067:
The ubiquitous
Prouhet-Thue-Morse sequence by Jean-Paul Allouche and Jeffrey Shallit
4470:
2951:
2891:
2665:
2304:
2297:
2136:
2042:
615:
606:
30:
2628:
It is also possible to draw the curve precisely using the following instructions:
4010:
4918:
4776:
4716:
4557:
4238:
Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009).
3630:
2003:
1869:
or eventually periodic (i.e., periodic after some initial nonperiodic segment).
39:
27:
Infinite binary sequence generated by repeated complementation and concatenation
4414:
3568:: by using its cube-free property (see above), he showed how to circumvent the
2827:
fairness of various tournament competitions, such as the kicking sequence of a
4922:
4613:
4328:
4278:. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge:
4189:
Barrow, John D. (2010). "Rowing and the Same-Sum
Problem Have Their Moments".
4078:
Fredricksen, Harold (1992). "Gray codes and the Thue-Morse-Hedlund sequence".
2946:
bits of the Thue–Morse sequence are mapped to 0 by a wide class of polynomial
2924:
2746:
2724:
159:
3552:
many times, not always by professional research mathematicians; for example,
3532:. However, Prouhet did not mention the sequence explicitly; this was left to
4860:
3597:
3533:
2769:
2312:
1966:(1) = 10: every 0 in a sequence is replaced with 01 and every 1 with 10. If
671:
83:
2139:
is algebraic over the field of rational functions, satisfying the equation
2864:). An ABBA serving pattern has also been found to improve the fairness of
2691:, in their discussion of how to fairly apportion a shared meal such as an
110:
When counting in binary, the digit sum modulo 2 is the Thue–Morse sequence
3553:
2905:
by giving earlier selections in each round to weaker teams. By contrast,
2779:
619:
4080:
Journal of
Combinatorial Mathematics and Combinatorial Computing (JCMCC)
3540:. The sequence was only brought to worldwide attention with the work of
4372:
Brlek, Srećko (1989). "Enumeration of Factors in the Thue-Morse Word".
4239:
4035:
3561:
2861:
2823:
1596:
are obtained by repeating one of these 4 strings, they all have length
98:
There are several equivalent ways of defining the Thue–Morse sequence.
4222:
2651:) = 1, move ahead by one unit, then rotate by an angle of π/3 radians.
2610:
2308:
1899:
be a word obtained by counting the ones between consecutive zeros in
2821:
proposed changing the sequential order to Thue–Morse to improve the
4052:
782:= 0110100110010110100101100110100110010110011010010110100110010110.
4532:
4453:
4319:
4241:
Combinatorics on words. Christoffel words and repetitions in words
4205:
3565:
666:
The Thue–Morse sequence in the form given above, as a sequence of
29:
4897:
4889:
4688:. Lecture Notes in Mathematics. Vol. 1794. Berlin, Germany:
4511:"How to Make the Most of a Shared Meal: Plan the Last Bite First"
4099:"On the Asymptotic Relative Change for Sequences of Permutations"
4139:
2857:
917:
Which can then be converted to a (reversed) string as follows:
2447:{\displaystyle \sum _{x\in S_{0}}x^{i}=\sum _{x\in S_{1}}x^{i}}
667:
58:
of the sequence obtained thus far. It is sometimes called the
4749:. Cambridge Tracts in Mathematics. Vol. 193. Cambridge:
4395:
Cohen-Zada, Danny; Krumer, Alex; Shapir, Offer Moshe (2018).
3789:
3777:
682:, then the next 2 elements must form the bitwise negation of
233:
that is different from the same bit in the representation of
229:, find the highest-order bit in the binary representation of
2798:
2699:
686:. Now we have defined the first 2 elements, and we recurse.
4911:
4900:
4892:
4878:
4631:"Tournaments, fairness and the Prouhet–Thue–Morse sequence"
4349:
The Win-Win Solution: Guaranteeing Fair Shares to Everybody
4244:. CRM Monograph Series. Vol. 27. Providence, RI, USA:
4011:"Where to Use and How not to Use Polynomial String Hashing"
3634:
4160:
Automatic Sequences: Theory, Applications, Generalizations
1861:. Notably, the Thue-Morse sequence is uniformly recurrent
2711:
on the interval and described their relationship to the
1549:{\displaystyle A{\overline {A}}AA{\overline {A}}A=010010}
4686:
Substitutions in dynamics, arithmetics and combinatorics
2022:; the only other fixed point is the bitwise negation of
304:"""Thue–Morse sequence."""
3801:
4397:"Testing the effect of serve order in tennis tiebreak"
3934:
2901:. Many professional sports leagues attempt to achieve
2782:, prompting a whimsical article in the popular press.
2125:{\displaystyle t(Z)=\sum _{n=0}^{\infty }T(n)Z^{n}\ .}
74:
of the Thue–Morse sequence. The full sequence begins:
4747:
Distribution modulo one and Diophantine approximation
4482:. Lecture Notes in Computer Science. Vol. 4036.
4147:
Matters Computational: Ideas, Algorithms, Source Code
3337:
3314:
3294:
3262:
3216:
2975:
2734:. As a consequence, the step function arising from
2374:
2296:) forms a subspace of the nonnegative integers under
2269:
2249:
2148:
2057:
1958:
is defined on alphabet {0,1} by the substitution map
1810:
1774:
1742:
1712:
1688:
1662:
1629:
1602:
1582:
1562:
1507:
1468:
1442:
1422:
1395:
1369:
1336:
1296:
1263:
1236:
1216:
1196:
1173:
1007:
463:
4908:. A technical application of the Thue–Morse Sequence
711:
Combining these, the first 8 elements are 01101001.
4780:
4346:
3635:"Sequence A010060 (Thue-Morse sequence)"
3501:
3320:
3300:
3288:term of the Thue-Morse sequence. In fact, for all
3280:
3248:
3199:
2446:
2288:
2255:
2223:
2124:
1816:
1792:
1760:
1724:
1694:
1674:
1648:
1615:
1588:
1568:
1548:
1493:
1454:
1428:
1408:
1381:
1355:
1322:
1282:
1249:
1222:
1202:
1182:
1127:
583:
2365:that have equal sums of powers up to k, that is:
4873:. (contains many applications and some history)
3922:
3910:
3657:
1576:starting at the 16th bit. Since all squares in
1323:{\displaystyle {\overline {A}}A{\overline {A}}}
705:Combining these, the first 4 elements are 0110.
2852:). As a result, ABBA is undergoing
2640:) = 0, rotate by an angle of π radians (180°),
337:# Note: assumes that (-1).bit_length() gives 1
4509:Levine, Lionel; Stange, Katherine E. (2012).
4402:Journal of Economic Behavior and Organization
3573:consecutive criterion can be evaded forever.
3564:, discovered it in 1929 in an application to
3517:The Thue–Morse sequence was first studied by
2679:taking turns taking turns taking turns . . .
2224:{\displaystyle Z+(1+Z)^{2}t+(1+Z)^{3}t^{2}=0}
699:Combining these, the first 2 elements are 01.
8:
4009:Pachocki, Jakub; Radoszewski, Jakub (2013).
3856:
3717:
2860:and English Federation professional soccer (
2327:can be defined as: given a positive integer
689:Spelling out the first few steps in detail:
610:Thue–Morse sequence generated by an L-System
4717:"Recursive Binary Sequences of Differences"
3899:
3844:
3593:First difference of the Thue–Morse sequence
3536:in 1906, who used it to found the study of
2894:to avoid wiggle on an eight-membered boat.
4871:The Ubiquitous Prouhet-Thue-Morse Sequence
4345:Brams, Steven J.; Taylor, Alan D. (1999).
4276:Combinatorics, automata, and number theory
3869:
3867:
3865:
3813:
3753:
4876:Thue–Morse Sequence over (1,2) (sequence
4531:
4452:
4318:
4204:
4051:
3895:
3893:
3641:On-Line Encyclopedia of Integer Sequences
3625:
3623:
3478:
3463:
3453:
3447:
3435:
3416:
3398:
3382:
3376:
3364:
3345:
3336:
3313:
3293:
3268:
3267:
3261:
3234:
3224:
3215:
3160:
3149:
3127:
3117:
3105:
3082:
3072:
3039:
3028:
3006:
2996:
2984:
2976:
2974:
2768:. An example showed how to pour cups of
2570:th element of an arithmetic progression.
2566:applied to the binomial representing the
2438:
2426:
2415:
2402:
2390:
2379:
2373:
2274:
2268:
2248:
2209:
2199:
2171:
2147:
2110:
2088:
2077:
2056:
1809:
1773:
1741:
1711:
1687:
1661:
1640:
1628:
1607:
1601:
1581:
1561:
1527:
1511:
1506:
1479:
1467:
1441:
1421:
1396:
1394:
1368:
1347:
1335:
1310:
1297:
1295:
1267:
1262:
1237:
1235:
1215:
1195:
1172:
1116:
1104:
1099:
1080:
1069:
1049:
1044:
1023:
1012:
1006:
568:
536:
519:
499:
472:
464:
462:
4353:. W. W. Norton & Co., Inc. pp.
3884:
3765:
3741:
3705:
3681:
624:
605:
444:The Thue–Morse sequence is the sequence
105:
4127:"How to pour the perfect cup of coffee"
3873:
3729:
3619:
2858:FIFA (European and World Championships)
2772:of equal strength from a carafe with a
662:Characterization using bitwise negation
4795:, Michael Waterman, Philippe Jacquet,
3946:
3935:Cohen-Zada, Krumer & Shapir (2018)
3653:
3651:
1842:(often much longer than the length of
1835:in the sequence, there is some length
1163:The Thue–Morse sequence contains many
436:(constant number of words) of memory.
4430:Cooper, Joshua; Dutle, Aaron (2013).
3693:
3669:
2845:), and 51% using full Thue–Morse (or
2617:The resulting curve converges to the
998:The sequence can also be defined by:
708:The bitwise negation of 0110 is 1001.
137:in this binary expansion is odd then
78:01101001100101101001011001101001....
7:
4542:10.4169/amer.math.monthly.119.07.550
4463:10.4169/amer.math.monthly.120.05.441
2897:Fairness is especially important in
618:: it is the output of the following
406:# Bit index is even, so toggle value
240:. If this bit is at an even index,
4923:"The Fairest Sharing Sequence Ever"
2810:, but for sequences of any length.
773:= 01101001100101101001011001101001.
4912:MusiNum - The Music in the Numbers
3826:Abel, Zachary (January 23, 2012).
3272:
3269:
2089:
1081:
1024:
259:, and otherwise it is the same as
25:
4665:Palacios-Huerta, Ignacio (2014).
4629:Palacios-Huerta, Ignacio (2012).
3249:{\displaystyle (t_{n})_{n\geq 0}}
2609:) = 1, rotate by an angle of π/3
1970:is the Thue–Morse sequence, then
1283:{\displaystyle A{\overline {A}}A}
702:The bitwise negation of 01 is 10.
192:) numbers, and numbers for which
4869:Allouche, J.-P.; Shallit, J. O.
4650:10.1111/j.1465-7295.2011.00435.x
4562:Algebraic combinatorics on words
4140:"1.16.4 The Thue–Morse sequence"
2660:In their book on the problem of
2319:The Prouhet–Tarry–Escott problem
4125:Abrahams, Marc (12 July 2010).
3828:"Thue-Morse Navigating Turtles"
3544:in 1921, when he applied it to
2719:functions. He showed that the
1409:{\displaystyle {\overline {A}}}
1250:{\displaystyle {\overline {A}}}
696:The bitwise negation of 0 is 1.
62:because of its applications to
4783:Applied combinatorics on words
4669:. Princeton University Press.
3493:
3487:
3428:
3409:
3357:
3338:
3231:
3217:
3187:
3181:
3066:
3060:
2727:can be expressed in terms of
2598:) = 0, move ahead by one unit,
2541:0 + 3 + 5 + 6 = 1 + 2 + 4 + 7.
2536:0 + 3 + 5 + 6 = 1 + 2 + 4 + 7,
2470:is a multiple of 2, given by:
2196:
2183:
2168:
2155:
2103:
2097:
2067:
2061:
1096:
1086:
594:for all non-negative integers
1:
4519:American Mathematical Monthly
4440:American Mathematical Monthly
4274:; Rigo, Michel, eds. (2010).
4246:American Mathematical Society
4097:Erickson, John (2018-10-30).
3985:"Third-Round Reversal Drafts"
2546:The condition requiring that
1827:The Thue–Morse sequence is a
614:The Thue–Morse sequence is a
144: = 1, if even then
4593:"When Thue-Morse meets Koch"
4388:10.1016/0166-218x(92)90274-e
4375:Discrete Applied Mathematics
3983:Allan, Ian (July 16, 2014).
3658:Allouche & Shallit (2003
3308:with real part greater than
3281:{\displaystyle n^{\rm {th}}}
2578:Fractals and turtle graphics
2325:Prouhet–Tarry–Escott problem
2235:In combinatorial game theory
1649:{\displaystyle 3\cdot 2^{n}}
1532:
1516:
1401:
1315:
1302:
1272:
1242:
82:The sequence is named after
4839:Encyclopedia of Mathematics
4192:American Journal of Physics
3608:Prouhet–Thue–Morse constant
3528:in 1851, who applied it to
2879:is the only arrangement of
2331:and a non-negative integer
1416:is the bitwise negation of
188:(intended to be similar to
48:Prouhet–Thue–Morse sequence
18:Prouhet-Thue-Morse sequence
4977:
4956:Fixed points (mathematics)
4801:Cambridge University Press
4751:Cambridge University Press
4415:10.1016/j.jebo.2017.12.012
4280:Cambridge University Press
4164:Cambridge University Press
3857:Levine & Stange (2012)
3631:Sloane, N. J. A.
2925:pick-up game of basketball
2881:port- and starboard-rowing
1831:: given any finite string
1167:: instances of the string
1151:th element if we start at
434:logarithmic number of bits
151: = 0. That is,
4614:10.1142/S0218348X05002908
4329:10.4310/JOC.2016.v7.n1.a5
3900:Cooper & Dutle (2013)
3845:Brams & Taylor (1999)
2504:consists of the integers
2480:consists of the integers
1494:{\displaystyle A=T_{0}=0}
4715:Richman, Robert (2001).
4307:Journal of Combinatorics
4018:Olympiads in Informatics
3814:Ma & Holdener (2005)
3754:Berthé & Rigo (2010)
3603:Komornik–Loreti constant
3550:discovered independently
3548:. The sequence has been
2907:fantasy football leagues
2554:th powers of any set of
1829:uniformly recurrent word
919:
795:
280:
211:Fast sequence generation
4149:. Springer. p. 44.
2819:Ignacio Palacios-Huerta
2466:This has a solution if
2289:{\displaystyle t_{n}=0}
1675:{\displaystyle n\geq 0}
1382:{\displaystyle k\geq 0}
1356:{\displaystyle A=T_{k}}
674:using the operation of
4745:Bugeaud, Yann (2012).
3923:Palacios-Huerta (2014)
3911:Palacios-Huerta (2012)
3538:combinatorics on words
3503:
3322:
3302:
3282:
3250:
3201:
2838:), 54% using ABBA (or
2560:arithmetic progression
2448:
2290:
2257:
2225:
2126:
2093:
1818:
1794:
1762:
1726:
1696:
1676:
1650:
1617:
1590:
1570:
1550:
1495:
1456:
1430:
1410:
1383:
1357:
1324:
1284:
1251:
1224:
1204:
1184:
1129:
1085:
1028:
611:
585:
111:
35:
4856:"Thue-Morse Sequence"
4834:"Thue-Morse sequence"
4667:Beautiful Game Theory
4432:"Greedy Galois Games"
4154:Allouche, Jean-Paul;
3960:"Fantasy Draft Types"
3832:Three-Cornered Things
3546:differential geometry
3504:
3323:
3303:
3283:
3251:
3202:
2962:Riemann zeta function
2449:
2291:
2258:
2226:
2127:
2073:
1819:
1795:
1793:{\displaystyle 1X1X1}
1763:
1761:{\displaystyle 0X0X0}
1727:
1697:
1677:
1651:
1618:
1616:{\displaystyle 2^{n}}
1591:
1571:
1551:
1496:
1457:
1431:
1411:
1384:
1358:
1325:
1285:
1252:
1225:
1205:
1185:
1130:
1065:
1008:
609:
586:
225:, and then, for each
199: = 0 to be
184: = 1 to be
109:
33:
4961:Parity (mathematics)
4797:Wojciech Szpankowski
4486:. pp. 280–291.
4138:Arndt, Jörg (2011).
3972:on October 12, 2018.
3802:Bolker et al. (2016)
3790:Berstel et al. (2009
3778:Berstel et al. (2009
3570:threefold repetition
3335:
3312:
3292:
3260:
3214:
2973:
2954:, which can lead to
2890:is one of only four
2675:balanced alternation
2656:Equitable sequencing
2372:
2307:). For the game of
2267:
2247:
2146:
2055:
2000:prolongable morphism
1808:
1772:
1740:
1732:. There are also no
1710:
1686:
1660:
1627:
1600:
1580:
1560:
1505:
1466:
1440:
1420:
1393:
1367:
1334:
1294:
1261:
1234:
1214:
1194:
1171:
1005:
461:
268: − 1
256: − 1
238: − 1
4215:2010AmJPh..78..728B
2762:continuous function
2689:Katherine E. Stange
2047:formal power series
2014:is essentially the
1932:overlapping squares
1890:. Furthermore, let
1734:overlapping squares
1725:{\displaystyle XXX}
1455:{\displaystyle k=0}
1436:. For instance, if
1210:denotes the string
764:= 0110100110010110.
655:(0 → 01), (1 → 10)
453:recurrence relation
440:Recurrence relation
125:, write the number
60:fair share sequence
4853:Weisstein, Eric W.
3718:Pytheas Fogg (2002
3644:. OEIS Foundation.
3499:
3446:
3375:
3318:
3298:
3278:
3246:
3197:
3195:
3116:
2995:
2903:competitive parity
2870:competitive rowing
2444:
2433:
2397:
2286:
2253:
2221:
2122:
2028:automatic sequence
1814:
1790:
1758:
1722:
1692:
1672:
1646:
1613:
1586:
1566:
1546:
1491:
1452:
1426:
1406:
1379:
1353:
1320:
1280:
1247:
1220:
1200:
1183:{\displaystyle XX}
1180:
1125:
620:Lindenmayer system
612:
581:
579:
112:
56:Boolean complement
36:
4810:978-0-521-84802-2
4760:978-0-521-11169-0
4699:978-3-540-44141-0
4571:978-0-521-18071-9
4493:978-3-540-35428-4
4364:978-0-393-04729-5
4289:978-0-521-51597-9
4255:978-0-8218-4480-9
4223:10.1119/1.3318808
4173:978-0-521-82332-6
3558:chess grandmaster
3469:
3431:
3404:
3360:
3321:{\displaystyle 1}
3301:{\displaystyle s}
3166:
3101:
3092:
3045:
2980:
2866:tennis tie-breaks
2829:penalty shoot-out
2523:For example, for
2454:for all integers
2411:
2375:
2256:{\displaystyle n}
2118:
2035:generating series
1930:does not contain
1923:= 2102012. Since
1817:{\displaystyle T}
1802:critical exponent
1695:{\displaystyle T}
1589:{\displaystyle T}
1569:{\displaystyle T}
1535:
1519:
1429:{\displaystyle A}
1404:
1318:
1305:
1275:
1245:
1223:{\displaystyle A}
1203:{\displaystyle X}
670:, can be defined
659:
658:
286:generate_sequence
173:. deemed numbers
102:Direct definition
16:(Redirected from
4968:
4951:Binary sequences
4937:
4935:
4933:
4927:
4899:
4891:
4881:
4866:
4865:
4847:
4822:
4786:
4772:
4733:
4721:
4711:
4680:
4661:
4638:Economic Inquiry
4635:
4625:
4597:
4583:
4553:
4535:
4515:
4505:
4474:
4456:
4436:
4426:
4391:
4368:
4352:
4340:
4322:
4301:
4267:
4234:
4208:
4185:
4156:Shallit, Jeffrey
4150:
4144:
4134:
4111:
4108:
4106:
4105:
4094:
4088:
4087:
4075:
4069:
4064:
4058:
4057:
4055:
4032:
4026:
4025:
4015:
4006:
4000:
3999:
3997:
3995:
3980:
3974:
3973:
3968:. Archived from
3956:
3950:
3944:
3938:
3932:
3926:
3920:
3914:
3908:
3902:
3897:
3888:
3882:
3876:
3871:
3860:
3854:
3848:
3842:
3836:
3835:
3823:
3817:
3811:
3805:
3799:
3793:
3787:
3781:
3775:
3769:
3763:
3757:
3751:
3745:
3739:
3733:
3727:
3721:
3715:
3709:
3703:
3697:
3691:
3685:
3679:
3673:
3667:
3661:
3655:
3646:
3645:
3627:
3583:Dejean's theorem
3560:and mathematics
3527:
3508:
3506:
3505:
3500:
3483:
3482:
3470:
3468:
3467:
3458:
3457:
3448:
3445:
3421:
3420:
3405:
3403:
3402:
3393:
3392:
3377:
3374:
3350:
3349:
3327:
3325:
3324:
3319:
3307:
3305:
3304:
3299:
3287:
3285:
3284:
3279:
3277:
3276:
3275:
3255:
3253:
3252:
3247:
3245:
3244:
3229:
3228:
3206:
3204:
3203:
3198:
3196:
3167:
3165:
3164:
3155:
3154:
3153:
3138:
3137:
3118:
3115:
3093:
3088:
3087:
3086:
3073:
3046:
3044:
3043:
3034:
3033:
3032:
3017:
3016:
2997:
2994:
2945:
2854:extensive trials
2693:Ethiopian dinner
2564:binomial theorem
2542:
2537:
2453:
2451:
2450:
2445:
2443:
2442:
2432:
2431:
2430:
2407:
2406:
2396:
2395:
2394:
2295:
2293:
2292:
2287:
2279:
2278:
2262:
2260:
2259:
2254:
2230:
2228:
2227:
2222:
2214:
2213:
2204:
2203:
2176:
2175:
2131:
2129:
2128:
2123:
2116:
2115:
2114:
2092:
2087:
2010:as fixed point:
1943:squarefree words
1941:are palindromic
1909:. For instance,
1854:block of length
1823:
1821:
1820:
1815:
1799:
1797:
1796:
1791:
1767:
1765:
1764:
1759:
1731:
1729:
1728:
1723:
1701:
1699:
1698:
1693:
1681:
1679:
1678:
1673:
1655:
1653:
1652:
1647:
1645:
1644:
1622:
1620:
1619:
1614:
1612:
1611:
1595:
1593:
1592:
1587:
1575:
1573:
1572:
1567:
1555:
1553:
1552:
1547:
1536:
1528:
1520:
1512:
1500:
1498:
1497:
1492:
1484:
1483:
1461:
1459:
1458:
1453:
1435:
1433:
1432:
1427:
1415:
1413:
1412:
1407:
1405:
1397:
1388:
1386:
1385:
1380:
1362:
1360:
1359:
1354:
1352:
1351:
1329:
1327:
1326:
1321:
1319:
1311:
1306:
1298:
1289:
1287:
1286:
1281:
1276:
1268:
1256:
1254:
1253:
1248:
1246:
1238:
1229:
1227:
1226:
1221:
1209:
1207:
1206:
1201:
1189:
1187:
1186:
1181:
1134:
1132:
1131:
1126:
1121:
1120:
1111:
1110:
1109:
1108:
1084:
1079:
1061:
1057:
1056:
1055:
1054:
1053:
1027:
1022:
994:Infinite product
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
913:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
693:We start with 0.
676:bitwise negation
625:
590:
588:
587:
582:
580:
573:
572:
550:
549:
524:
523:
507:
506:
477:
476:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
398:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
270:
258:
239:
224:
21:
4976:
4975:
4971:
4970:
4969:
4967:
4966:
4965:
4941:
4940:
4931:
4929:
4925:
4917:
4877:
4851:
4850:
4832:
4829:
4811:
4775:
4761:
4744:
4741:
4739:Further reading
4736:
4725:Complex Systems
4719:
4714:
4700:
4690:Springer-Verlag
4683:
4677:
4664:
4633:
4628:
4595:
4586:
4572:
4556:
4513:
4508:
4494:
4484:Springer-Verlag
4477:
4434:
4429:
4394:
4371:
4365:
4344:
4304:
4290:
4272:Berthé, Valérie
4270:
4256:
4237:
4188:
4174:
4153:
4142:
4137:
4124:
4120:
4115:
4114:
4103:
4101:
4096:
4095:
4091:
4077:
4076:
4072:
4065:
4061:
4034:
4033:
4029:
4013:
4008:
4007:
4003:
3993:
3991:
3982:
3981:
3977:
3958:
3957:
3953:
3945:
3941:
3933:
3929:
3921:
3917:
3909:
3905:
3898:
3891:
3885:Abrahams (2010)
3883:
3879:
3872:
3863:
3855:
3851:
3843:
3839:
3825:
3824:
3820:
3812:
3808:
3800:
3796:
3788:
3784:
3776:
3772:
3764:
3760:
3752:
3748:
3740:
3736:
3728:
3724:
3716:
3712:
3704:
3700:
3692:
3688:
3680:
3676:
3668:
3664:
3656:
3649:
3629:
3628:
3621:
3616:
3588:Fabius function
3579:
3521:
3515:
3474:
3459:
3449:
3412:
3394:
3378:
3341:
3333:
3332:
3310:
3309:
3290:
3289:
3263:
3258:
3257:
3230:
3220:
3212:
3211:
3194:
3193:
3168:
3156:
3145:
3123:
3119:
3098:
3097:
3078:
3074:
3047:
3035:
3024:
3002:
2998:
2971:
2970:
2964:
2956:hash collisions
2943:
2940:
2938:Hash collisions
2933:
2922:
2915:
2889:
2878:
2851:
2844:
2837:
2803:
2739:
2732:
2704:
2658:
2584:turtle graphics
2580:
2540:
2535:
2517:
2503:
2493:
2479:
2434:
2422:
2398:
2386:
2370:
2369:
2364:
2357:
2343:= { 0, 1, ...,
2321:
2270:
2265:
2264:
2245:
2244:
2237:
2205:
2195:
2167:
2144:
2143:
2106:
2053:
2052:
2018:fixed point of
1994:. The morphism
1939:
1928:
1922:
1915:
1908:
1898:
1881:
1859:
1840:
1806:
1805:
1770:
1769:
1738:
1737:
1736:: instances of
1708:
1707:
1706:: instances of
1684:
1683:
1658:
1657:
1636:
1625:
1624:
1603:
1598:
1597:
1578:
1577:
1558:
1557:
1503:
1502:
1475:
1464:
1463:
1438:
1437:
1418:
1417:
1391:
1390:
1365:
1364:
1343:
1332:
1331:
1292:
1291:
1259:
1258:
1232:
1231:
1212:
1211:
1192:
1191:
1169:
1168:
1161:
1146:
1112:
1100:
1095:
1045:
1040:
1033:
1029:
1003:
1002:
996:
991:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
946:thue_morse_bits
945:
942:
939:
936:
933:
930:
927:
924:
921:
915:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
801:thue_morse_bits
800:
797:
781:
772:
763:
754:
745:
736:
727:
664:
604:
578:
577:
564:
551:
532:
529:
528:
515:
508:
495:
492:
491:
478:
468:
459:
458:
451:satisfying the
449:
442:
430:
429:
426:
423:
420:
417:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
269:
260:
257:
248:
245:
234:
222:
216:
213:
197:
182:
160:even parity bit
156:
149:
142:
123:
114:To compute the
104:
96:
68:parity sequence
52:binary sequence
28:
23:
22:
15:
12:
11:
5:
4974:
4972:
4964:
4963:
4958:
4953:
4943:
4942:
4939:
4938:
4928:. standupmaths
4915:
4909:
4903:
4895:
4887:
4874:
4867:
4848:
4828:
4827:External links
4825:
4824:
4823:
4809:
4793:Sophie Schbath
4789:Gesine Reinert
4773:
4759:
4740:
4737:
4735:
4734:
4712:
4698:
4681:
4676:978-0691144023
4675:
4662:
4644:(3): 848–849.
4626:
4608:(3): 191–206.
4589:Holdener, Judy
4584:
4570:
4554:
4526:(7): 550–565.
4506:
4492:
4475:
4447:(5): 441–451.
4427:
4392:
4382:(1–3): 83–96.
4369:
4363:
4342:
4313:(1): 117–133.
4302:
4288:
4268:
4254:
4235:
4199:(7): 728–732.
4186:
4172:
4151:
4135:
4121:
4119:
4116:
4113:
4112:
4089:
4070:
4059:
4046:(article 98).
4027:
4001:
3975:
3951:
3939:
3927:
3915:
3903:
3889:
3877:
3874:Richman (2001)
3861:
3849:
3837:
3818:
3806:
3794:
3782:
3770:
3766:Lothaire (2011
3758:
3746:
3742:Lothaire (2011
3734:
3730:Krieger (2006)
3722:
3720:, p. 103)
3710:
3708:, p. 113)
3706:Lothaire (2011
3698:
3686:
3682:Lothaire (2011
3674:
3662:
3647:
3618:
3617:
3615:
3612:
3611:
3610:
3605:
3600:
3595:
3590:
3585:
3578:
3575:
3519:Eugène Prouhet
3514:
3511:
3510:
3509:
3498:
3495:
3492:
3489:
3486:
3481:
3477:
3473:
3466:
3462:
3456:
3452:
3444:
3441:
3438:
3434:
3430:
3427:
3424:
3419:
3415:
3411:
3408:
3401:
3397:
3391:
3388:
3385:
3381:
3373:
3370:
3367:
3363:
3359:
3356:
3353:
3348:
3344:
3340:
3317:
3297:
3274:
3271:
3266:
3243:
3240:
3237:
3233:
3227:
3223:
3219:
3208:
3207:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3169:
3163:
3159:
3152:
3148:
3144:
3141:
3136:
3133:
3130:
3126:
3122:
3114:
3111:
3108:
3104:
3100:
3099:
3096:
3091:
3085:
3081:
3077:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3048:
3042:
3038:
3031:
3027:
3023:
3020:
3015:
3012:
3009:
3005:
3001:
2993:
2990:
2987:
2983:
2979:
2978:
2963:
2960:
2948:hash functions
2939:
2936:
2931:
2920:
2913:
2887:
2876:
2849:
2842:
2835:
2801:
2737:
2730:
2709:step functions
2702:
2657:
2654:
2653:
2652:
2641:
2615:
2614:
2599:
2579:
2576:
2544:
2543:
2538:
2521:
2520:
2515:
2501:
2496:
2491:
2477:
2464:
2463:
2441:
2437:
2429:
2425:
2421:
2418:
2414:
2410:
2405:
2401:
2393:
2389:
2385:
2382:
2378:
2362:
2355:
2347:-1 } into two
2320:
2317:
2285:
2282:
2277:
2273:
2252:
2236:
2233:
2232:
2231:
2220:
2217:
2212:
2208:
2202:
2198:
2194:
2191:
2188:
2185:
2182:
2179:
2174:
2170:
2166:
2163:
2160:
2157:
2154:
2151:
2133:
2132:
2121:
2113:
2109:
2105:
2102:
2099:
2096:
2091:
2086:
2083:
2080:
2076:
2072:
2069:
2066:
2063:
2060:
1937:
1926:
1920:
1913:
1903:
1894:
1876:
1857:
1838:
1813:
1789:
1786:
1783:
1780:
1777:
1757:
1754:
1751:
1748:
1745:
1721:
1718:
1715:
1691:
1671:
1668:
1665:
1643:
1639:
1635:
1632:
1610:
1606:
1585:
1565:
1545:
1542:
1539:
1534:
1531:
1526:
1523:
1518:
1515:
1510:
1490:
1487:
1482:
1478:
1474:
1471:
1451:
1448:
1445:
1425:
1403:
1400:
1378:
1375:
1372:
1350:
1346:
1342:
1339:
1317:
1314:
1309:
1304:
1301:
1279:
1274:
1271:
1266:
1244:
1241:
1219:
1199:
1179:
1176:
1160:
1157:
1142:
1136:
1135:
1124:
1119:
1115:
1107:
1103:
1098:
1094:
1091:
1088:
1083:
1078:
1075:
1072:
1068:
1064:
1060:
1052:
1048:
1043:
1039:
1036:
1032:
1026:
1021:
1018:
1015:
1011:
995:
992:
920:
796:
787:
786:
783:
779:
774:
770:
765:
761:
756:
752:
747:
743:
738:
734:
729:
725:
716:
715:
712:
709:
706:
703:
700:
697:
694:
663:
660:
657:
656:
653:
649:
648:
645:
641:
640:
637:
633:
632:
629:
603:
600:
592:
591:
576:
571:
567:
563:
560:
557:
554:
552:
548:
545:
542:
539:
535:
531:
530:
527:
522:
518:
514:
511:
509:
505:
502:
498:
494:
493:
490:
487:
484:
481:
479:
475:
471:
467:
466:
447:
441:
438:
281:
264:
252:
243:
223: = 0
220:
212:
209:
195:
180:
168:John H. Conway
154:
147:
140:
135:number of ones
121:
103:
100:
95:
92:
80:
79:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4973:
4962:
4959:
4957:
4954:
4952:
4949:
4948:
4946:
4924:
4920:
4916:
4913:
4910:
4907:
4904:
4902:
4896:
4894:
4888:
4885:
4880:
4875:
4872:
4868:
4863:
4862:
4857:
4854:
4849:
4845:
4841:
4840:
4835:
4831:
4830:
4826:
4820:
4816:
4812:
4806:
4802:
4798:
4794:
4790:
4785:
4784:
4778:
4774:
4770:
4766:
4762:
4756:
4752:
4748:
4743:
4742:
4738:
4732:(4): 381–392.
4731:
4727:
4726:
4718:
4713:
4709:
4705:
4701:
4695:
4691:
4687:
4682:
4678:
4672:
4668:
4663:
4659:
4655:
4651:
4647:
4643:
4639:
4632:
4627:
4623:
4619:
4615:
4611:
4607:
4603:
4602:
4594:
4590:
4585:
4581:
4577:
4573:
4567:
4563:
4559:
4555:
4551:
4547:
4543:
4539:
4534:
4529:
4525:
4521:
4520:
4512:
4507:
4503:
4499:
4495:
4489:
4485:
4481:
4476:
4472:
4468:
4464:
4460:
4455:
4450:
4446:
4442:
4441:
4433:
4428:
4424:
4420:
4416:
4412:
4408:
4404:
4403:
4398:
4393:
4389:
4385:
4381:
4377:
4376:
4370:
4366:
4360:
4356:
4351:
4350:
4343:
4338:
4334:
4330:
4326:
4321:
4316:
4312:
4308:
4303:
4299:
4295:
4291:
4285:
4282:. p. 7.
4281:
4277:
4273:
4269:
4265:
4261:
4257:
4251:
4247:
4243:
4242:
4236:
4232:
4228:
4224:
4220:
4216:
4212:
4207:
4202:
4198:
4194:
4193:
4187:
4183:
4179:
4175:
4169:
4165:
4161:
4157:
4152:
4148:
4141:
4136:
4132:
4128:
4123:
4122:
4117:
4110:
4100:
4093:
4090:
4085:
4081:
4074:
4071:
4068:
4063:
4060:
4054:
4049:
4045:
4041:
4037:
4031:
4028:
4023:
4019:
4012:
4005:
4002:
3990:
3989:Fantasy Index
3986:
3979:
3976:
3971:
3967:
3966:
3961:
3955:
3952:
3948:
3947:Barrow (2010)
3943:
3940:
3936:
3931:
3928:
3924:
3919:
3916:
3912:
3907:
3904:
3901:
3896:
3894:
3890:
3886:
3881:
3878:
3875:
3870:
3868:
3866:
3862:
3858:
3853:
3850:
3846:
3841:
3838:
3833:
3829:
3822:
3819:
3815:
3810:
3807:
3803:
3798:
3795:
3792:, p. 80)
3791:
3786:
3783:
3780:, p. 70)
3779:
3774:
3771:
3768:, p. 31)
3767:
3762:
3759:
3755:
3750:
3747:
3744:, p. 30)
3743:
3738:
3735:
3731:
3726:
3723:
3719:
3714:
3711:
3707:
3702:
3699:
3695:
3690:
3687:
3684:, p. 11)
3683:
3678:
3675:
3671:
3666:
3663:
3660:, p. 15)
3659:
3654:
3652:
3648:
3643:
3642:
3636:
3632:
3626:
3624:
3620:
3613:
3609:
3606:
3604:
3601:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
3580:
3576:
3574:
3571:
3567:
3563:
3559:
3555:
3551:
3547:
3543:
3542:Marston Morse
3539:
3535:
3531:
3530:number theory
3525:
3520:
3512:
3496:
3490:
3484:
3479:
3475:
3471:
3464:
3460:
3454:
3450:
3442:
3439:
3436:
3432:
3425:
3422:
3417:
3413:
3406:
3399:
3395:
3389:
3386:
3383:
3379:
3371:
3368:
3365:
3361:
3354:
3351:
3346:
3342:
3331:
3330:
3329:
3315:
3295:
3264:
3241:
3238:
3235:
3225:
3221:
3190:
3184:
3178:
3175:
3172:
3170:
3161:
3157:
3150:
3146:
3142:
3139:
3134:
3131:
3128:
3124:
3120:
3112:
3109:
3106:
3102:
3094:
3089:
3083:
3079:
3075:
3069:
3063:
3057:
3054:
3051:
3049:
3040:
3036:
3029:
3025:
3021:
3018:
3013:
3010:
3007:
3003:
2999:
2991:
2988:
2985:
2981:
2969:
2968:
2967:
2961:
2959:
2957:
2953:
2949:
2937:
2935:
2930:
2926:
2919:
2912:
2908:
2904:
2900:
2899:player drafts
2895:
2893:
2886:
2882:
2875:
2871:
2867:
2863:
2859:
2855:
2848:
2841:
2834:
2830:
2826:
2825:
2820:
2815:
2811:
2809:
2805:
2804:
2795:
2793:
2788:
2783:
2781:
2778:
2777:concentration
2775:
2771:
2767:
2763:
2759:
2758:monotonically
2755:
2752:
2748:
2744:
2740:
2733:
2726:
2722:
2718:
2714:
2710:
2706:
2705:
2696:
2694:
2690:
2686:
2685:Lionel Levine
2682:
2680:
2676:
2671:
2667:
2663:
2662:fair division
2655:
2650:
2646:
2642:
2639:
2635:
2631:
2630:
2629:
2626:
2624:
2623:fractal curve
2620:
2612:
2608:
2604:
2600:
2597:
2593:
2589:
2588:
2587:
2585:
2577:
2575:
2571:
2569:
2565:
2561:
2557:
2553:
2549:
2539:
2534:
2533:
2532:
2530:
2526:
2518:
2511:
2507:
2500:
2497:
2494:
2487:
2483:
2476:
2473:
2472:
2471:
2469:
2461:
2457:
2439:
2435:
2427:
2423:
2419:
2416:
2412:
2408:
2403:
2399:
2391:
2387:
2383:
2380:
2376:
2368:
2367:
2366:
2361:
2354:
2350:
2346:
2342:
2338:
2334:
2330:
2326:
2318:
2316:
2314:
2310:
2306:
2303:
2299:
2283:
2280:
2275:
2271:
2250:
2242:
2234:
2218:
2215:
2210:
2206:
2200:
2192:
2189:
2186:
2180:
2177:
2172:
2164:
2161:
2158:
2152:
2149:
2142:
2141:
2140:
2138:
2119:
2111:
2107:
2100:
2094:
2084:
2081:
2078:
2074:
2070:
2064:
2058:
2051:
2050:
2049:
2048:
2044:
2040:
2036:
2031:
2029:
2025:
2021:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1954:
1953:
1946:
1944:
1940:
1933:
1929:
1919:
1912:
1907:
1902:
1897:
1893:
1889:
1885:
1880:
1875:
1872:The sequence
1870:
1868:
1865:being either
1864:
1860:
1853:
1849:
1845:
1841:
1834:
1830:
1825:
1811:
1803:
1787:
1784:
1781:
1778:
1775:
1755:
1752:
1749:
1746:
1743:
1735:
1719:
1716:
1713:
1705:
1689:
1669:
1666:
1663:
1641:
1637:
1633:
1630:
1608:
1604:
1583:
1563:
1543:
1540:
1537:
1529:
1524:
1521:
1513:
1508:
1501:. The square
1488:
1485:
1480:
1476:
1472:
1469:
1449:
1446:
1443:
1423:
1398:
1376:
1373:
1370:
1348:
1344:
1340:
1337:
1312:
1307:
1299:
1277:
1269:
1264:
1239:
1217:
1197:
1177:
1174:
1166:
1158:
1156:
1154:
1150:
1145:
1141:
1122:
1117:
1113:
1105:
1101:
1092:
1089:
1076:
1073:
1070:
1066:
1062:
1058:
1050:
1046:
1041:
1037:
1034:
1030:
1019:
1016:
1013:
1009:
1001:
1000:
999:
993:
918:
794:
792:
784:
778:
775:
769:
766:
760:
757:
751:
748:
742:
739:
733:
730:
724:
721:
720:
719:
713:
710:
707:
704:
701:
698:
695:
692:
691:
690:
687:
685:
681:
677:
673:
669:
661:
654:
651:
650:
646:
643:
642:
638:
635:
634:
630:
627:
626:
623:
621:
617:
608:
601:
599:
597:
574:
569:
565:
561:
558:
555:
553:
546:
543:
540:
537:
533:
525:
520:
516:
512:
510:
503:
500:
496:
488:
485:
482:
480:
473:
469:
457:
456:
455:
454:
450:
439:
437:
435:
279:
277:
272:
267:
263:
255:
251:
247:differs from
246:
237:
232:
228:
219:
210:
208:
206:
202:
198:
191:
187:
183:
176:
172:
169:
165:
161:
157:
150:
143:
136:
132:
128:
124:
117:
108:
101:
99:
93:
91:
89:
88:Marston Morse
85:
77:
76:
75:
73:
69:
65:
64:fair division
61:
57:
53:
49:
45:
41:
32:
19:
4930:. Retrieved
4919:Parker, Matt
4859:
4837:
4782:
4777:Lothaire, M.
4746:
4729:
4723:
4685:
4666:
4641:
4637:
4605:
4599:
4561:
4558:Lothaire, M.
4523:
4517:
4479:
4444:
4438:
4406:
4400:
4379:
4373:
4348:
4310:
4306:
4275:
4240:
4196:
4190:
4159:
4146:
4131:The Guardian
4130:
4102:. Retrieved
4092:
4083:
4079:
4073:
4062:
4043:
4039:
4036:Tóth, László
4030:
4021:
4017:
4004:
3994:September 1,
3992:. Retrieved
3988:
3978:
3970:the original
3963:
3954:
3942:
3930:
3918:
3906:
3880:
3852:
3840:
3831:
3821:
3809:
3797:
3785:
3773:
3761:
3749:
3737:
3725:
3713:
3701:
3694:Brlek (1989)
3689:
3677:
3670:Arndt (2011)
3665:
3638:
3516:
3209:
2965:
2952:power of two
2942:The initial
2941:
2928:
2917:
2910:
2896:
2884:
2873:
2846:
2839:
2832:
2822:
2816:
2812:
2807:
2799:
2791:
2784:
2753:
2735:
2728:
2720:
2700:
2697:
2683:
2678:
2674:
2666:Steven Brams
2659:
2648:
2644:
2637:
2633:
2627:
2616:
2606:
2602:
2595:
2591:
2581:
2572:
2567:
2555:
2551:
2547:
2545:
2528:
2524:
2522:
2513:
2509:
2505:
2498:
2489:
2485:
2481:
2474:
2467:
2465:
2459:
2455:
2359:
2352:
2344:
2340:
2332:
2328:
2322:
2305:exclusive or
2298:nim-addition
2241:evil numbers
2240:
2238:
2137:power series
2134:
2043:binary field
2038:
2034:
2032:
2023:
2019:
2015:
2011:
2007:
1995:
1991:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1949:
1947:
1935:
1934:, the words
1931:
1924:
1917:
1910:
1905:
1900:
1895:
1891:
1887:
1878:
1873:
1871:
1862:
1855:
1851:
1847:
1846:) such that
1843:
1836:
1832:
1826:
1733:
1703:
1702:contains no
1164:
1162:
1152:
1148:
1143:
1139:
1137:
997:
916:
788:
776:
767:
758:
749:
740:
731:
722:
717:
688:
683:
679:
665:
616:morphic word
613:
595:
593:
445:
443:
431:
273:
265:
261:
253:
249:
241:
235:
230:
226:
217:
214:
204:
203:(similar to
200:
193:
189:
185:
178:
174:
170:
163:
152:
145:
138:
126:
119:
115:
113:
97:
81:
67:
59:
47:
43:
37:
4409:: 106–115.
3522: [
2794:probability
2760:decreasing
2747:polynomials
2670:Alan Taylor
2558:numbers in
2239:The set of
2006:{0,1} with
2004:free monoid
1988:fixed point
1950:Thue–Morse
1850:appears in
1556:appears in
755:= 01101001.
672:recursively
207:) numbers.
177:satisfying
118:th element
40:mathematics
4945:Categories
4932:20 January
4819:1133.68067
4769:1260.11001
4708:1014.11015
4580:1221.68183
4502:1227.68074
4298:1197.68006
4264:1161.68043
4182:1086.11015
4118:References
4104:2021-01-31
4053:2211.13570
3328:, we have
2806:of length
2743:orthogonal
2725:derivative
2717:Rademacher
2619:Koch curve
2512:for which
2488:for which
2458:from 1 to
2313:nim-values
1978:) is also
1962:(0) = 01,
1884:palindrome
1159:Properties
785:And so on.
714:And so on.
636:Constants
628:Variables
373:bit_length
331:seq_length
292:seq_length
133:. If the
94:Definition
44:Thue–Morse
4861:MathWorld
4844:EMS Press
4587:Ma, Jun;
4533:1104.0961
4454:1110.1137
4337:118040795
4320:1304.6756
4231:119207447
4206:0911.3551
4024:: 90–100.
3598:Gray code
3534:Axel Thue
3485:ζ
3440:≥
3433:∑
3423:−
3387:−
3369:≥
3362:∑
3239:≥
3179:ζ
3132:−
3110:≥
3103:∑
3080:π
3058:ζ
3011:−
2989:≥
2982:∑
2950:modulo a
2927:mirrors
2774:nonlinear
2420:∈
2413:∑
2384:∈
2377:∑
2337:partition
2243:(numbers
2090:∞
2075:∑
2041:over the
1667:≥
1656:for some
1634:⋅
1533:¯
1517:¯
1402:¯
1374:≥
1363:for some
1316:¯
1303:¯
1273:¯
1243:¯
1090:−
1082:∞
1067:∑
1038:−
1025:∞
1010:∏
562:−
84:Axel Thue
4779:(2005).
4658:54036493
4601:Fractals
4591:(2005).
4560:(2011).
4550:14537479
4423:89610106
4158:(2003).
4040:Integers
3577:See also
3554:Max Euwe
2792:a priori
2780:gradient
2527:= 8 and
2351:subsets
2349:disjoint
2339:the set
1982:. Thus,
1952:morphism
1916:= 2 and
1886:for any
1867:periodic
1330:, where
1190:, where
970:<<
900:<<
891:<<
867:<<
858:<<
602:L-system
72:prefixes
4926:(video)
4882:in the
4879:A001285
4846:, 2001
4622:2166279
4471:1291901
4211:Bibcode
3965:NFL.com
3633:(ed.).
3562:teacher
3513:History
3256:is the
2862:EFL Cup
2824:ex post
2766:flatter
2611:radians
2311:, evil
2302:bitwise
2045:is the
2002:on the
1863:without
1462:, then
1165:squares
1147:is the
746:= 0110.
158:is the
50:is the
4817:
4807:
4767:
4757:
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4696:
4673:
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4180:
4170:
3210:where
2787:Galois
2770:coffee
2582:Using
2309:Kayles
2117:
1824:is 2.
1800:. The
1544:010010
1138:where
985:"
940:"
909:return
791:Python
652:Rules
644:Start
276:Python
186:odious
131:binary
42:, the
4720:(PDF)
4654:S2CID
4634:(PDF)
4596:(PDF)
4546:S2CID
4528:arXiv
4514:(PDF)
4467:S2CID
4449:arXiv
4435:(PDF)
4419:S2CID
4355:36–44
4333:S2CID
4315:arXiv
4227:S2CID
4201:arXiv
4143:(PDF)
4048:arXiv
4014:(PDF)
3614:Notes
3566:chess
3526:]
2868:. In
2751:order
2713:Walsh
2677:, or
2613:(60°)
2531:= 2,
2263:with
2135:This
1998:is a
1986:is a
1882:is a
1852:every
1704:cubes
1290:, or
1155:= 0.
931:print
834:range
737:= 01.
639:None
631:0, 1
427:value
424:yield
421:value
409:value
391:&
325:range
307:value
171:et al
4934:2016
4898:OEIS
4890:OEIS
4884:OEIS
4805:ISBN
4755:ISBN
4694:ISBN
4671:ISBN
4566:ISBN
4488:ISBN
4359:ISBN
4284:ISBN
4250:ISBN
4168:ISBN
3996:2020
3639:The
3556:, a
2892:rigs
2715:and
2687:and
2668:and
2621:, a
2519:= 1.
2495:= 0,
2358:and
2323:The
2033:The
2016:only
1948:The
1389:and
912:bits
885:bits
846:bits
816:bits
728:= 0.
668:bits
205:even
201:evil
162:for
86:and
4815:Zbl
4765:Zbl
4704:Zbl
4646:doi
4610:doi
4576:Zbl
4538:doi
4524:119
4498:Zbl
4459:doi
4445:120
4411:doi
4407:146
4384:doi
4325:doi
4294:Zbl
4260:Zbl
4219:doi
4178:Zbl
2856:in
2749:of
2745:to
2741:is
2723:th
2707:as
2643:If
2632:If
2601:If
2590:If
2508:in
2484:in
2037:of
1990:of
1804:of
1768:or
1623:or
825:for
798:def
789:In
718:So
316:for
298:int
283:def
274:In
271:.
190:odd
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66:or
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2030:.
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810:):
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3052:=
3041:2
3037:n
3030:n
3026:t
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3008:n
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3000:5
2992:1
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2808:2
2802:n
2800:T
2754:n
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2731:n
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2721:n
2703:n
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2649:n
2647:(
2645:t
2638:n
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2634:t
2607:n
2605:(
2603:t
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2594:(
2592:t
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2468:N
2462:.
2460:k
2456:i
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2436:x
2428:1
2424:S
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2409:=
2404:i
2400:x
2392:0
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2381:x
2363:1
2360:S
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2353:S
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2329:N
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2251:n
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2207:t
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2187:1
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2079:n
2071:=
2068:)
2065:Z
2062:(
2059:t
2039:T
2024:T
2020:μ
2012:T
2008:T
1996:μ
1992:μ
1984:T
1980:T
1976:T
1974:(
1972:μ
1968:T
1964:μ
1960:μ
1956:μ
1938:n
1936:q
1927:n
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1921:2
1918:q
1914:1
1911:q
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1833:X
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1788:1
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1714:X
1690:T
1670:0
1664:n
1642:n
1638:2
1631:3
1609:n
1605:2
1584:T
1564:T
1541:=
1538:A
1530:A
1525:A
1522:A
1514:A
1509:A
1489:0
1486:=
1481:0
1477:T
1473:=
1470:A
1450:0
1447:=
1444:k
1424:A
1399:A
1377:0
1371:k
1349:k
1345:T
1341:=
1338:A
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1270:A
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1198:X
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