3589:
2260:
2304:
time. Apostolico and
Preparata give an algorithm using suffix trees. Crochemore uses partitioning in his algorithm. Main and Lorentz provide an algorithm based on the divide-and-conquer method. A naive implementation may require
2117:
3904:
1075:
1623:
2679:
2542:
3979:
2968:. A morphism is called square-free if the image of every square-free word is square-free. A morphism is called k–square-free if the image of every square-free word of length k is square-free.
4019:
2828:
469:
1658:
1471:
507:
Since there are infinitely many square-free words over three-letter alphabets, this implies there are also infinitely many square-free words over an alphabet with more than three letters.
963:
1436:
437:
1529:
1384:
906:
154:
3721:
1566:
1500:
792:
688:
3488:
3375:
3329:
1174:
1150:
1126:
1102:
840:
816:
720:
659:
3539:
2421:
2028:
1951:
2302:
1818:
1722:
3247:
2746:
1982:
1752:
221:
3575:
2340:
1903:
1869:
3758:
2861:
1269:
753:
489:
104:
1299:
3931:
2925:
2890:
2775:
2106:
1337:
1228:
3409:
3282:
3192:
3152:
3112:
3006:
340:
251:
3812:
3060:
174:
3782:
1784:
1689:
4615:. Encyclopedia of Mathematics and Its Applications. Vol. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.).
3500:
Over a ternary alphabet, there are exactly 144 uniform square-free morphisms of rank 11 and no uniform square-free morphisms with a lower rank than 11.
2688:
2067:
Every (k+1)-ary square-free word can be the output of
Algorithm R2F, because on each iteration it can append any letter except for the last letter of
2547:
one forms a new sequence in which each term is the difference of two consecutive terms of the Thue–Morse sequence. The resulting square-free word is
4659:
4624:
4597:
4563:
4430:
3821:
3649:
Note that over a more than three-letter alphabet there are square-free words of any length without an arbitrary three-letter combination.
968:
3631:
Note that over a more than three-letter alphabet there are square-free words of any length without an arbitrary two-letter combination.
3580:
Note that, if a morphism over a ternary alphabet is not uniform, then this morphism is square-free if and only if it is 5-square-free.
4332:
4110:
2255:{\displaystyle N=n\left(1+{\frac {2}{k^{2}}}+{\frac {1}{k^{3}}}+{\frac {4}{k^{4}}}+O\left({\frac {1}{k^{5}}}\right)\right)+O(1).}
4691:
1587:
2553:
3639:
Over a ternary alphabet, a square-free word of length more than 36 contains all the square-free three-letter combinations.
4555:
3605:
Over a ternary alphabet, a square-free word of length more than 13 contains all the square-free two-letter combinations.
2437:
3951:
3991:
2786:
4616:
4589:
4356:
4285:
Main, Michael G; Lorentz, Richard J (Sep 1984). "An O(n log n) algorithm for finding all repetitions in a string".
2707:
2696:
1628:
1441:
442:
4686:
911:
36:
1389:
391:
2428:
2373:
1505:
845:
109:
3668:
1542:
1476:
1354:
758:
664:
4643:
4489:
2379:
One example of an infinite square-free word over an alphabet of size 3 is the word over the alphabet
3434:
3336:
3290:
1193:
1155:
1131:
1107:
1083:
821:
797:
701:
640:
3518:
4450:
4376:
3013:
2382:
1991:
1908:
2269:
3208:
2713:
1960:
1790:
1730:
1694:
188:
4655:
4620:
4593:
4559:
4524:
4474:
4426:
4338:
4328:
4302:
4267:
4232:
4148:
4106:
3544:
2309:
1838:
492:
52:
3726:
2833:
1881:
1248:
725:
474:
77:
4665:
4630:
4569:
4516:
4418:
4384:
4368:
4294:
4259:
4250:
Crochemore, Max (Oct 1981). "An optimal algorithm for computing the repetitions in a word".
4222:
4189:
4179:
4138:
4098:
4075:
2424:
2262:
Note that there exists an algorithm that can verify the square-freeness of a word of length
1278:
3909:
2903:
2868:
2753:
2079:
1310:
1201:
1192:
over any alphabet with three or more letters. This algorithm is based on a modification of
4669:
4651:
4634:
4573:
4388:
3385:
3258:
3168:
3128:
3088:
2982:
496:
316:
227:
3787:
3608:
This can be proved by constructing a square-free word without the two-letter combination
3039:
2949:. It is possible to prove that the sequence converges to the infinite square-free word
499:. The lower bound can be found by finding a substitution that preserves square-freeness.
159:
4493:
3642:
However, there are square-free words of any length without the three-letter combination
3588:
3767:
3511:
to it. The resulting words preserve the property of square-freeness. For example, let
1760:
1665:
4449:
Zolotov, Boris (2015). "Another
Solution to the Thue Problem of Non-Repeating Words".
4680:
4475:"The minimal density of a letter in an infinite ternary square-free word is 883/3215"
4380:
4298:
4263:
4227:
4210:
4143:
4126:
28:
4327:. Départements de mathématiques et d'informatique, Université du Québec à Montréal.
4545:
4406:
224:, there are infinitely many square-free words. It is possible to count the number
4102:
4079:
3503:
To obtain an infinite square-free words, start with any square-free word such as
2074:
The expected number of random k-ary letters used by
Algorithm R2F to construct a
4608:
4581:
17:
4520:
4372:
4184:
4167:
1128:
is square-free. A computer search shows that there are no 2-dimensional words
4528:
4342:
4306:
4271:
4236:
4152:
514:-ary square-free words, rounded off to 7 digits after the decimal point, for
4507:
Ochem, Pascal (2007). "Letter frequency in infinite repetition-free words".
4422:
2360:
51:
is not empty. Thus, a square-free word can also be defined as a word that
2700:
2356:
72:
4549:
4194:
3899:{\displaystyle \liminf _{l\to \infty }{\frac {|w_{l}|_{a}}{|w_{l}|}}}
1196:: it randomly selects letters from a k-letter alphabet to generate a
4548:; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009).
4455:
2975:
is square-free if and only if it is 3-square-free. In other words,
1188:(random-t(w)o-free) that can generate a square-free word of length
1070:{\displaystyle t\geq 0,x_{t}=w_{{i_{1}}+{j_{1}t},{i_{2}}+{j_{2}t}}}
4551:
Combinatorics on words. Christoffel words and repetitions in words
4066:
Shur, Arseny (2011). "Growth properties of power-free languages".
4646:; Ferenczi, SĂ©bastien; Mauduit, Christian; Siegel, Anne (eds.).
3012:
of length 3. It is possible to find a square-free morphism by
39:(a sequence of symbols) that does not contain any squares. A
4411:
Annual
Symposium on Theoretical Aspects of Computer Science
4037:
2683:
4650:. Lecture Notes in Mathematics. Vol. 1794. Berlin:
4648:
Substitutions in dynamics, arithmetics and combinatorics
4325:
Axel Thue's papers on repetitions in words a translation
3624:
is the longest square-free word without the combination
4211:"Optimal off-line detection of repetitions in a string"
4093:
Berthe, Valerie; Rigo, Michel, eds. (2016), "Preface",
2954:
0121021201210120210201202120102101201021202102012021...
2343:
time to verify the square-freeness of a word of length
1618:{\displaystyle \color {gray}w=136263163\in \Sigma _{6}}
510:
The following table shows the exact growth rate of the
4554:. CRM Monograph Series. Vol. 27. Providence, RI:
2674:{\displaystyle 1,0,-1,1,-1,0,1,0,-1,0,1,-1,1,0,-1,...}
1884:
1793:
1697:
1357:
445:
3994:
3954:
3912:
3824:
3790:
3770:
3729:
3671:
3547:
3521:
3437:
3388:
3339:
3293:
3261:
3211:
3171:
3131:
3091:
3042:
3029:
a square-free morphism with the lowest possible rank
2985:
2906:
2871:
2836:
2789:
2756:
2716:
2556:
2440:
2385:
2355:
There exist infinitely long square-free words in any
2312:
2272:
2120:
2082:
1994:
1963:
1911:
1841:
1763:
1733:
1668:
1631:
1590:
1545:
1508:
1479:
1444:
1392:
1313:
1281:
1251:
1204:
1158:
1134:
1110:
1086:
971:
914:
848:
824:
800:
761:
728:
704:
667:
643:
477:
394:
319:
230:
191:
162:
112:
80:
3988:
in an infinite ternary square-free word is equal to
3948:
in an infinite ternary square-free word is equal to
1080:
Carpi proves that there exists a 2-dimensional word
261:
The number of ternary square-free words of length n
3814:is the length of the word. The density of a letter
4013:
3973:
3925:
3898:
3806:
3776:
3752:
3715:
3569:
3533:
3482:
3403:
3369:
3323:
3276:
3241:
3186:
3146:
3106:
3054:
3000:
2919:
2884:
2855:
2822:
2769:
2740:
2673:
2537:{\displaystyle 0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0...}
2536:
2415:
2334:
2296:
2254:
2100:
2022:
1976:
1945:
1897:
1863:
1812:
1778:
1746:
1716:
1683:
1652:
1617:
1560:
1523:
1494:
1465:
1430:
1378:
1331:
1293:
1263:
1222:
1168:
1152:over a 7-letter alphabet, such that every line of
1144:
1120:
1104:over a 16-letter alphabet such that every line of
1096:
1069:
957:
900:
834:
810:
786:
747:
714:
682:
653:
483:
463:
431:
334:
245:
215:
168:
148:
98:
4097:, Cambridge University Press, pp. xi–xviii,
3974:{\displaystyle {\frac {883}{3215}}\approx 0.2747}
2779:be any square-free word starting with the letter
4014:{\displaystyle {\frac {255}{653}}\approx 0.3905}
3826:
3507:, and successively apply a square-free morphism
106:, the only square-free words are the empty word
2964:Infinite square-free words can be generated by
2823:{\displaystyle \{w_{i}\mid i\in \mathbb {N} \}}
4127:"Multidimensional unrepetitive configurations"
464:{\textstyle 1.3017597<\alpha <1.3017619}
1653:{\displaystyle \color {gray}\chi _{w}=361245}
1576:is to the right of the rightmost position of
1466:{\displaystyle \color {gray}w\in \Sigma _{k}}
8:
4209:Apostolico, A.; Preparata, F.P. (Feb 1983).
3078:the list of all square-free words of length
2817:
2790:
2735:
2717:
2410:
2386:
1424:
1394:
210:
192:
93:
81:
958:{\displaystyle {\text{gcd}}(j_{1},j_{2})=1}
4168:"Generating square-free words efficiently"
4095:Combinatorics, Words and Symbolic Dynamics
2971:Crochemore proves that a uniform morphism
2359:with three or more letters, as proved by
1431:{\displaystyle \color {gray}\{1,...,k+1\}}
4454:
4407:"Some Recent Results on Squarefree Words"
4359:(1957). "A problem on strings of beads".
4226:
4193:
4183:
4142:
3995:
3993:
3955:
3953:
3917:
3911:
3888:
3882:
3873:
3865:
3860:
3853:
3844:
3841:
3829:
3823:
3799:
3791:
3789:
3769:
3744:
3739:
3730:
3728:
3705:
3697:
3689:
3684:
3675:
3672:
3670:
3552:
3546:
3520:
3436:
3387:
3338:
3292:
3260:
3210:
3170:
3130:
3090:
3041:
2984:
2911:
2905:
2876:
2870:
2841:
2835:
2813:
2812:
2797:
2788:
2761:
2755:
2715:
2710:is defined recursively over the alphabet
2555:
2439:
2384:
2323:
2311:
2271:
2217:
2208:
2190:
2181:
2170:
2161:
2150:
2141:
2119:
2081:
1999:
1993:
1968:
1962:
1922:
1910:
1889:
1883:
1850:
1842:
1840:
1798:
1792:
1762:
1738:
1732:
1702:
1696:
1667:
1637:
1630:
1608:
1589:
1551:
1544:
1514:
1507:
1485:
1478:
1456:
1443:
1391:
1363:
1356:
1312:
1280:
1250:
1203:
1160:
1159:
1157:
1136:
1135:
1133:
1112:
1111:
1109:
1088:
1087:
1085:
1055:
1050:
1040:
1035:
1022:
1017:
1007:
1002:
1001:
988:
970:
940:
927:
915:
913:
892:
879:
866:
853:
847:
826:
825:
823:
802:
801:
799:
763:
762:
760:
733:
727:
706:
705:
703:
674:
670:
669:
666:
645:
644:
642:
476:
444:
432:{\displaystyle c(n)=\Theta (\alpha ^{n})}
420:
393:
318:
229:
190:
161:
111:
79:
3587:
3584:Letter combinations in square-free words
2431:. That is, from the Thue–Morse sequence
1524:{\displaystyle \color {gray}\Sigma _{k}}
520:
259:
4029:
3255:from the current loop (advance to next
1379:{\textstyle \color {gray}\Sigma _{k+1}}
901:{\displaystyle i_{1},i_{2},j_{1},j_{2}}
253:of ternary square-free words of length
149:{\displaystyle \epsilon ,0,1,01,10,010}
3716:{\displaystyle {\frac {|w|_{a}}{|w|}}}
3592:Extending a square-free word to avoid
1561:{\displaystyle \color {gray}\chi _{w}}
1495:{\displaystyle \color {gray}\chi _{w}}
4468:
4466:
4444:
4442:
4413:. Lecture Notes in Computer Science.
4400:
4398:
2960:Generating infinite square-free words
503:Alphabet with more than three letters
7:
4318:
4316:
4061:
4059:
4057:
1568:if the right most position of
722:is called a 2-dimensional word. Let
3515:be a square-free morphism, then as
3008:is square-free for all square-free
1905:uniformly at random append
1180:Generating finite square-free words
1161:
1137:
1113:
1089:
827:
803:
764:
707:
646:
3836:
3528:
1988:elements to the right and setting
1886:
1795:
1699:
1632:
1605:
1591:
1546:
1511:
1509:
1480:
1453:
1445:
1393:
1360:
1358:
1184:Shur proposes an algorithm called
787:{\displaystyle {\textbf {w}}(m,n)}
410:
25:
3577:is an infinite square-free word.
2830:recursively as follows: the word
1787:followed by all other letters of
4613:Algebraic combinatorics on words
3984:The maximum density of a letter
3944:The minimal density of a letter
3760:is the number of occurrences of
3082:over a ternary alphabet
2110:-ary square-free word of length
683:{\displaystyle \mathbb {N} ^{2}}
3635:Avoid three-letter combinations
3628:and its length is equal to 13.
4252:Information Processing Letters
3889:
3874:
3861:
3845:
3833:
3800:
3792:
3740:
3731:
3706:
3698:
3685:
3676:
3564:
3558:
3525:
3483:{\displaystyle h(0),h(1),h(2)}
3477:
3471:
3462:
3456:
3447:
3441:
3398:
3392:
3364:
3358:
3349:
3343:
3318:
3312:
3303:
3297:
3271:
3265:
3236:
3230:
3221:
3215:
3181:
3175:
3141:
3135:
3101:
3095:
2995:
2989:
2979:is square-free if and only if
2329:
2316:
2291:
2276:
2246:
2240:
2095:
2083:
2011:
2005:
1940:
1928:
1851:
1843:
1773:
1767:
1678:
1672:
1326:
1314:
1217:
1205:
946:
920:
781:
769:
426:
413:
404:
398:
329:
323:
240:
234:
1:
4556:American Mathematical Society
3601:Avoid two-letter combinations
3370:{\displaystyle h(2)\neq h(0)}
3324:{\displaystyle h(0)\neq h(1)}
1386:is the alphabet with letters
1169:{\displaystyle {\textbf {w}}}
1145:{\displaystyle {\textbf {w}}}
1121:{\displaystyle {\textbf {w}}}
1097:{\displaystyle {\textbf {w}}}
835:{\displaystyle {\textbf {w}}}
811:{\displaystyle {\textbf {x}}}
715:{\displaystyle {\textbf {w}}}
654:{\displaystyle {\textbf {w}}}
4509:Theoretical Computer Science
4482:Journal of Integer Sequences
4405:Berstel, Jean (April 1984).
4299:10.1016/0196-6774(84)90021-x
4264:10.1016/0020-0190(81)90024-7
4228:10.1016/0304-3975(83)90109-3
4215:Theoretical Computer Science
4172:Theoretical Computer Science
4144:10.1016/0304-3975(88)90080-1
4131:Theoretical Computer Science
4103:10.1017/cbo9781139924733.001
4080:10.1016/j.cosrev.2012.09.001
3534:{\displaystyle w\to \infty }
2416:{\displaystyle \{-1,0,+1\}}
2044:ends with a square of rank
2023:{\displaystyle \chi _{w}=a}
1946:{\displaystyle a=\chi _{w}}
518:in the range from 4 to 15:
4708:
4617:Cambridge University Press
4590:Cambridge University Press
4473:Khalyavin, Andrey (2007).
3933:is the prefix of the word
2351:Infinite square-free words
2297:{\displaystyle O(n\log n)}
1813:{\textstyle \Sigma _{k+1}}
1717:{\textstyle \Sigma _{k+1}}
388:This number is bounded by
4642:Pytheas Fogg, N. (2002).
4521:10.1016/j.tcs.2007.03.027
4373:10.1017/S0025557200236115
4185:10.1016/j.tcs.2015.07.027
3242:{\displaystyle h(1)=h(2)}
2741:{\displaystyle \{0,1,2\}}
2706:Another example found by
1977:{\displaystyle \chi _{w}}
1747:{\displaystyle \chi _{w}}
1724:uniformly at random
216:{\displaystyle \{0,1,2\}}
3657:The density of a letter
3570:{\displaystyle h^{w}(0)}
2695:
2372:First difference of the
2335:{\displaystyle O(n^{2})}
1898:{\textstyle \Sigma _{k}}
1864:{\displaystyle |w|<n}
1820:in increasing order
184:Over a ternary alphabet
62:Finite square-free words
4423:10.1007/3-540-12920-0_2
4068:Computer Science Review
3818:in an infinite word is
3753:{\displaystyle |w|_{a}}
2856:{\displaystyle w_{i+1}}
2423:obtained by taking the
1264:{\displaystyle k\geq 2}
1232:-ary square-free word.
748:{\displaystyle w_{m,n}}
526:-ary square-free words
484:{\displaystyle \alpha }
99:{\displaystyle \{0,1\}}
4692:Combinatorics on words
4586:Combinatorics on words
4323:Berstel, Jean (1994).
4125:Carpi, Arturo (1988).
4015:
3975:
3927:
3900:
3808:
3778:
3754:
3717:
3597:
3571:
3535:
3484:
3405:
3371:
3325:
3284:)
3278:
3243:
3188:
3148:
3108:
3056:
3002:
2921:
2886:
2857:
2824:
2771:
2742:
2675:
2538:
2417:
2336:
2298:
2256:
2102:
2024:
1978:
1947:
1899:
1865:
1814:
1780:
1748:
1718:
1685:
1654:
1619:
1562:
1525:
1502:is the permutation of
1496:
1467:
1432:
1380:
1341:-ary square-free word
1333:
1295:
1294:{\displaystyle n>1}
1265:
1224:
1170:
1146:
1122:
1098:
1071:
959:
902:
836:
812:
788:
749:
716:
684:
655:
485:
465:
433:
336:
247:
217:
170:
150:
100:
43:is a word of the form
4287:Journal of Algorithms
4166:Shur, Arseny (2015).
4016:
3976:
3928:
3926:{\displaystyle w_{l}}
3901:
3809:
3779:
3755:
3718:
3591:
3572:
3536:
3485:
3406:
3372:
3326:
3279:
3244:
3189:
3149:
3109:
3057:
3022:square-free_morphism
3003:
2922:
2920:{\displaystyle w_{i}}
2887:
2885:{\displaystyle w_{i}}
2858:
2825:
2772:
2770:{\displaystyle w_{1}}
2743:
2676:
2539:
2418:
2337:
2299:
2257:
2103:
2101:{\displaystyle (k+1)}
2025:
1979:
1948:
1900:
1866:
1815:
1781:
1749:
1719:
1686:
1655:
1620:
1584:. For example,
1563:
1526:
1497:
1468:
1433:
1381:
1334:
1332:{\displaystyle (k+1)}
1296:
1266:
1225:
1223:{\displaystyle (k+1)}
1171:
1147:
1123:
1099:
1072:
960:
903:
837:
813:
789:
750:
717:
685:
656:
495:and approximation by
486:
471:. The upper bound on
466:
434:
337:
248:
218:
171:
151:
101:
3992:
3952:
3910:
3822:
3788:
3768:
3727:
3669:
3545:
3519:
3435:
3404:{\displaystyle h(w)}
3386:
3337:
3291:
3277:{\displaystyle h(1)}
3259:
3209:
3187:{\displaystyle h(2)}
3169:
3147:{\displaystyle h(1)}
3129:
3107:{\displaystyle h(0)}
3089:
3040:
3001:{\displaystyle h(w)}
2983:
2966:square-free morphism
2904:
2869:
2834:
2787:
2754:
2714:
2554:
2438:
2383:
2310:
2270:
2118:
2080:
1992:
1961:
1909:
1882:
1839:
1791:
1761:
1731:
1695:
1666:
1629:
1588:
1543:
1506:
1477:
1442:
1390:
1355:
1311:
1279:
1249:
1202:
1156:
1132:
1108:
1084:
969:
912:
846:
822:
798:
759:
726:
702:
698:is an alphabet and
665:
641:
475:
443:
392:
335:{\displaystyle c(n)}
317:
246:{\displaystyle c(n)}
228:
189:
160:
110:
78:
4494:2007JIntS..10...65K
3807:{\displaystyle |w|}
3653:Density of a letter
3055:{\displaystyle k=3}
2783:. Define the words
2429:Thue–Morse sequence
2374:Thue–Morse sequence
1984:shifting the first
1438:.) (For a word
1194:entropy compression
633:2-dimensional words
527:
522:Growth rate of the
262:
169:{\displaystyle 101}
4011:
3971:
3923:
3896:
3840:
3804:
3774:
3750:
3713:
3598:
3567:
3531:
3480:
3401:
3367:
3321:
3274:
3239:
3184:
3144:
3104:
3052:
3014:brute-force search
2998:
2917:
2894:by replacing each
2882:
2853:
2820:
2767:
2738:
2671:
2534:
2413:
2332:
2294:
2252:
2098:
2020:
1974:
1943:
1895:
1861:
1810:
1776:
1744:
1714:
1681:
1650:
1649:
1615:
1614:
1558:
1557:
1521:
1520:
1492:
1491:
1463:
1462:
1428:
1427:
1376:
1375:
1329:
1291:
1261:
1220:
1166:
1142:
1118:
1094:
1067:
955:
898:
832:
808:
784:
745:
712:
680:
651:
521:
481:
461:
429:
332:
260:
243:
213:
166:
146:
96:
53:avoids the pattern
4661:978-3-540-44141-0
4626:978-0-521-18071-9
4599:978-0-521-59924-5
4565:978-0-8218-4480-9
4432:978-3-540-12920-2
4003:
3963:
3894:
3825:
3777:{\displaystyle w}
3711:
3661:in a finite word
2863:is obtained from
2223:
2196:
2176:
2156:
1779:{\displaystyle w}
1684:{\displaystyle w}
1163:
1139:
1115:
1091:
918:
829:
805:
766:
709:
648:
630:
629:
491:can be found via
386:
385:
16:(Redirected from
4699:
4687:Formal languages
4673:
4638:
4603:
4577:
4533:
4532:
4504:
4498:
4497:
4479:
4470:
4461:
4460:
4458:
4446:
4437:
4436:
4402:
4393:
4392:
4353:
4347:
4346:
4320:
4311:
4310:
4282:
4276:
4275:
4247:
4241:
4240:
4230:
4206:
4200:
4199:
4197:
4187:
4163:
4157:
4156:
4146:
4122:
4116:
4115:
4090:
4084:
4083:
4063:
4052:
4051:
4049:
4048:
4038:"A006156 - OEIS"
4034:
4020:
4018:
4017:
4012:
4004:
3996:
3987:
3980:
3978:
3977:
3972:
3964:
3956:
3947:
3940:
3936:
3932:
3930:
3929:
3924:
3922:
3921:
3905:
3903:
3902:
3897:
3895:
3893:
3892:
3887:
3886:
3877:
3871:
3870:
3869:
3864:
3858:
3857:
3848:
3842:
3839:
3817:
3813:
3811:
3810:
3805:
3803:
3795:
3783:
3781:
3780:
3775:
3763:
3759:
3757:
3756:
3751:
3749:
3748:
3743:
3734:
3722:
3720:
3719:
3714:
3712:
3710:
3709:
3701:
3695:
3694:
3693:
3688:
3679:
3673:
3664:
3660:
3645:
3627:
3622:
3619:
3616:
3611:
3595:
3576:
3574:
3573:
3568:
3557:
3556:
3540:
3538:
3537:
3532:
3514:
3510:
3506:
3497:
3493:
3489:
3487:
3486:
3481:
3425:
3421:
3418:all square-free
3410:
3408:
3407:
3402:
3376:
3374:
3373:
3368:
3330:
3328:
3327:
3322:
3283:
3281:
3280:
3275:
3248:
3246:
3245:
3240:
3193:
3191:
3190:
3185:
3153:
3151:
3150:
3145:
3113:
3111:
3110:
3105:
3081:
3061:
3059:
3058:
3053:
3032:
3011:
3007:
3005:
3004:
2999:
2978:
2974:
2955:
2948:
2944:
2940:
2936:
2932:
2928:
2926:
2924:
2923:
2918:
2916:
2915:
2897:
2893:
2891:
2889:
2888:
2883:
2881:
2880:
2862:
2860:
2859:
2854:
2852:
2851:
2829:
2827:
2826:
2821:
2816:
2802:
2801:
2782:
2778:
2776:
2774:
2773:
2768:
2766:
2765:
2747:
2745:
2744:
2739:
2686:
2680:
2678:
2677:
2672:
2543:
2541:
2540:
2535:
2425:first difference
2422:
2420:
2419:
2414:
2346:
2341:
2339:
2338:
2333:
2328:
2327:
2303:
2301:
2300:
2295:
2265:
2261:
2259:
2258:
2253:
2233:
2229:
2228:
2224:
2222:
2221:
2209:
2197:
2195:
2194:
2182:
2177:
2175:
2174:
2162:
2157:
2155:
2154:
2142:
2113:
2109:
2107:
2105:
2104:
2099:
2070:
2064:
2058:
2054:
2051:delete the last
2047:
2043:
2037:
2033:
2029:
2027:
2026:
2021:
2004:
2003:
1987:
1983:
1981:
1980:
1975:
1973:
1972:
1956:
1952:
1950:
1949:
1944:
1927:
1926:
1904:
1902:
1901:
1896:
1894:
1893:
1877:
1870:
1868:
1867:
1862:
1854:
1846:
1827:
1819:
1817:
1816:
1811:
1809:
1808:
1785:
1783:
1782:
1777:
1753:
1751:
1750:
1745:
1743:
1742:
1723:
1721:
1720:
1715:
1713:
1712:
1690:
1688:
1687:
1682:
1661:
1659:
1657:
1656:
1651:
1642:
1641:
1624:
1622:
1621:
1616:
1613:
1612:
1583:
1579:
1575:
1571:
1567:
1565:
1564:
1559:
1556:
1555:
1538:
1534:
1530:
1528:
1527:
1522:
1519:
1518:
1501:
1499:
1498:
1493:
1490:
1489:
1472:
1470:
1469:
1464:
1461:
1460:
1437:
1435:
1434:
1429:
1385:
1383:
1382:
1377:
1374:
1373:
1348:
1344:
1340:
1338:
1336:
1335:
1330:
1300:
1298:
1297:
1292:
1270:
1268:
1267:
1262:
1231:
1229:
1227:
1226:
1221:
1191:
1176:is square-free.
1175:
1173:
1172:
1167:
1165:
1164:
1151:
1149:
1148:
1143:
1141:
1140:
1127:
1125:
1124:
1119:
1117:
1116:
1103:
1101:
1100:
1095:
1093:
1092:
1076:
1074:
1073:
1068:
1066:
1065:
1064:
1060:
1059:
1046:
1045:
1044:
1031:
1027:
1026:
1013:
1012:
1011:
993:
992:
964:
962:
961:
956:
945:
944:
932:
931:
919:
916:
907:
905:
904:
899:
897:
896:
884:
883:
871:
870:
858:
857:
842:if there exists
841:
839:
838:
833:
831:
830:
817:
815:
814:
809:
807:
806:
793:
791:
790:
785:
768:
767:
754:
752:
751:
746:
744:
743:
721:
719:
718:
713:
711:
710:
697:
693:
689:
687:
686:
681:
679:
678:
673:
660:
658:
657:
652:
650:
649:
584:
534:
528:
525:
517:
513:
490:
488:
487:
482:
470:
468:
467:
462:
438:
436:
435:
430:
425:
424:
343:
341:
339:
338:
333:
268:
263:
256:
252:
250:
249:
244:
222:
220:
219:
214:
180:Ternary alphabet
175:
173:
172:
167:
155:
153:
152:
147:
105:
103:
102:
97:
57:
50:
46:
33:square-free word
21:
4707:
4706:
4702:
4701:
4700:
4698:
4697:
4696:
4677:
4676:
4662:
4652:Springer-Verlag
4644:Berthé, Valérie
4641:
4627:
4607:
4600:
4580:
4566:
4544:
4541:
4536:
4506:
4505:
4501:
4477:
4472:
4471:
4464:
4448:
4447:
4440:
4433:
4404:
4403:
4396:
4355:
4354:
4350:
4335:
4322:
4321:
4314:
4284:
4283:
4279:
4249:
4248:
4244:
4208:
4207:
4203:
4165:
4164:
4160:
4124:
4123:
4119:
4113:
4092:
4091:
4087:
4065:
4064:
4055:
4046:
4044:
4036:
4035:
4031:
4027:
3990:
3989:
3985:
3950:
3949:
3945:
3938:
3934:
3913:
3908:
3907:
3878:
3872:
3859:
3849:
3843:
3820:
3819:
3815:
3786:
3785:
3766:
3765:
3761:
3738:
3725:
3724:
3696:
3683:
3674:
3667:
3666:
3665:is defined as
3662:
3658:
3655:
3643:
3637:
3625:
3620:
3617:
3614:
3612:. As a result,
3609:
3603:
3593:
3586:
3548:
3543:
3542:
3517:
3516:
3512:
3508:
3504:
3498:
3495:
3491:
3433:
3432:
3423:
3419:
3384:
3383:
3335:
3334:
3289:
3288:
3257:
3256:
3207:
3206:
3167:
3166:
3127:
3126:
3087:
3086:
3079:
3038:
3037:
3030:
3009:
2981:
2980:
2976:
2972:
2962:
2953:
2946:
2942:
2938:
2934:
2930:
2907:
2902:
2901:
2899:
2895:
2872:
2867:
2866:
2864:
2837:
2832:
2831:
2793:
2785:
2784:
2780:
2757:
2752:
2751:
2749:
2712:
2711:
2704:
2682:
2552:
2551:
2436:
2435:
2381:
2380:
2377:
2369:
2353:
2344:
2319:
2308:
2307:
2268:
2267:
2263:
2213:
2204:
2186:
2166:
2146:
2134:
2130:
2116:
2115:
2111:
2078:
2077:
2075:
2068:
2065:
2062:
2056:
2052:
2045:
2041:
2035:
2031:
1995:
1990:
1989:
1985:
1964:
1959:
1958:
1954:
1918:
1907:
1906:
1885:
1880:
1879:
1875:
1837:
1836:
1825:
1794:
1789:
1788:
1759:
1758:
1734:
1729:
1728:
1698:
1693:
1692:
1664:
1663:
1633:
1627:
1626:
1604:
1586:
1585:
1581:
1577:
1573:
1569:
1547:
1541:
1540:
1536:
1532:
1510:
1504:
1503:
1481:
1475:
1474:
1452:
1440:
1439:
1388:
1387:
1359:
1353:
1352:
1350:
1346:
1342:
1309:
1308:
1306:
1277:
1276:
1247:
1246:
1200:
1199:
1197:
1189:
1182:
1154:
1153:
1130:
1129:
1106:
1105:
1082:
1081:
1051:
1036:
1018:
1003:
997:
984:
967:
966:
936:
923:
910:
909:
888:
875:
862:
849:
844:
843:
820:
819:
796:
795:
757:
756:
729:
724:
723:
700:
699:
695:
691:
668:
663:
662:
639:
638:
637:Consider a map
635:
582:
581:alphabet size (
532:
531:alphabet size (
523:
515:
511:
505:
473:
472:
441:
440:
416:
390:
389:
315:
314:
312:
266:
254:
226:
225:
187:
186:
182:
158:
157:
108:
107:
76:
75:
69:
67:Binary alphabet
64:
55:
48:
44:
23:
22:
18:Squarefree word
15:
12:
11:
5:
4705:
4703:
4695:
4694:
4689:
4679:
4678:
4675:
4674:
4660:
4639:
4625:
4605:
4598:
4578:
4564:
4540:
4537:
4535:
4534:
4515:(3): 388–392.
4499:
4462:
4438:
4431:
4394:
4348:
4334:978-2892761405
4333:
4312:
4293:(3): 422–432.
4277:
4258:(5): 244–250.
4242:
4221:(3): 297–315.
4201:
4158:
4137:(2): 233–241.
4117:
4111:
4085:
4074:(5–6): 28–43.
4053:
4028:
4026:
4023:
4010:
4007:
4002:
3999:
3970:
3967:
3962:
3959:
3920:
3916:
3891:
3885:
3881:
3876:
3868:
3863:
3856:
3852:
3847:
3838:
3835:
3832:
3828:
3827:lim inf
3802:
3798:
3794:
3773:
3747:
3742:
3737:
3733:
3708:
3704:
3700:
3692:
3687:
3682:
3678:
3654:
3651:
3636:
3633:
3602:
3599:
3585:
3582:
3566:
3563:
3560:
3555:
3551:
3530:
3527:
3524:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3400:
3397:
3394:
3391:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3273:
3270:
3267:
3264:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3183:
3180:
3177:
3174:
3143:
3140:
3137:
3134:
3103:
3100:
3097:
3094:
3051:
3048:
3045:
3018:
2997:
2994:
2991:
2988:
2961:
2958:
2957:
2956:
2914:
2910:
2879:
2875:
2850:
2847:
2844:
2840:
2819:
2815:
2811:
2808:
2805:
2800:
2796:
2792:
2764:
2760:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2703:
2694:
2693:
2692:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2545:
2544:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2376:
2370:
2368:
2365:
2352:
2349:
2331:
2326:
2322:
2318:
2315:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2251:
2248:
2245:
2242:
2239:
2236:
2232:
2227:
2220:
2216:
2212:
2207:
2203:
2200:
2193:
2189:
2185:
2180:
2173:
2169:
2165:
2160:
2153:
2149:
2145:
2140:
2137:
2133:
2129:
2126:
2123:
2097:
2094:
2091:
2088:
2085:
2019:
2016:
2013:
2010:
2007:
2002:
1998:
1971:
1967:
1953:to the end of
1942:
1939:
1936:
1933:
1930:
1925:
1921:
1917:
1914:
1892:
1888:
1860:
1857:
1853:
1849:
1845:
1828:of iterations
1807:
1804:
1801:
1797:
1775:
1772:
1769:
1766:
1741:
1737:
1711:
1708:
1705:
1701:
1680:
1677:
1674:
1671:
1648:
1645:
1640:
1636:
1611:
1607:
1603:
1600:
1597:
1594:
1554:
1550:
1517:
1513:
1488:
1484:
1459:
1455:
1451:
1448:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1372:
1369:
1366:
1362:
1328:
1325:
1322:
1319:
1316:
1290:
1287:
1284:
1260:
1257:
1254:
1245:alphabet size
1234:
1219:
1216:
1213:
1210:
1207:
1181:
1178:
1063:
1058:
1054:
1049:
1043:
1039:
1034:
1030:
1025:
1021:
1016:
1010:
1006:
1000:
996:
991:
987:
983:
980:
977:
974:
954:
951:
948:
943:
939:
935:
930:
926:
922:
895:
891:
887:
882:
878:
874:
869:
865:
861:
856:
852:
783:
780:
777:
774:
771:
742:
739:
736:
732:
677:
672:
634:
631:
628:
627:
624:
621:
618:
615:
612:
609:
605:
604:
601:
598:
595:
592:
589:
586:
578:
577:
574:
571:
568:
565:
562:
559:
555:
554:
551:
548:
545:
542:
539:
536:
504:
501:
493:Fekete's Lemma
480:
460:
457:
454:
451:
448:
428:
423:
419:
415:
412:
409:
406:
403:
400:
397:
384:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
331:
328:
325:
322:
309:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
242:
239:
236:
233:
212:
209:
206:
203:
200:
197:
194:
181:
178:
165:
145:
142:
139:
136:
133:
130:
127:
124:
121:
118:
115:
95:
92:
89:
86:
83:
71:Over a binary
68:
65:
63:
60:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4704:
4693:
4690:
4688:
4685:
4684:
4682:
4671:
4667:
4663:
4657:
4653:
4649:
4645:
4640:
4636:
4632:
4628:
4622:
4618:
4614:
4610:
4606:
4601:
4595:
4591:
4588:. Cambridge:
4587:
4583:
4579:
4575:
4571:
4567:
4561:
4557:
4553:
4552:
4547:
4546:Berstel, Jean
4543:
4542:
4538:
4530:
4526:
4522:
4518:
4514:
4510:
4503:
4500:
4495:
4491:
4487:
4483:
4476:
4469:
4467:
4463:
4457:
4452:
4445:
4443:
4439:
4434:
4428:
4424:
4420:
4416:
4412:
4408:
4401:
4399:
4395:
4390:
4386:
4382:
4378:
4374:
4370:
4366:
4362:
4358:
4352:
4349:
4344:
4340:
4336:
4330:
4326:
4319:
4317:
4313:
4308:
4304:
4300:
4296:
4292:
4288:
4281:
4278:
4273:
4269:
4265:
4261:
4257:
4253:
4246:
4243:
4238:
4234:
4229:
4224:
4220:
4216:
4212:
4205:
4202:
4196:
4191:
4186:
4181:
4177:
4173:
4169:
4162:
4159:
4154:
4150:
4145:
4140:
4136:
4132:
4128:
4121:
4118:
4114:
4112:9781139924733
4108:
4104:
4100:
4096:
4089:
4086:
4081:
4077:
4073:
4069:
4062:
4060:
4058:
4054:
4043:
4039:
4033:
4030:
4024:
4022:
4008:
4005:
4000:
3997:
3982:
3968:
3965:
3960:
3957:
3942:
3918:
3914:
3883:
3879:
3866:
3854:
3850:
3830:
3796:
3771:
3745:
3735:
3702:
3690:
3680:
3652:
3650:
3647:
3640:
3634:
3632:
3629:
3623:
3606:
3600:
3590:
3583:
3581:
3578:
3561:
3553:
3549:
3522:
3501:
3474:
3468:
3465:
3459:
3453:
3450:
3444:
3438:
3431:
3428:
3417:
3413:
3395:
3389:
3382:
3379:
3361:
3355:
3352:
3346:
3340:
3333:
3315:
3309:
3306:
3300:
3294:
3287:
3268:
3262:
3254:
3251:
3233:
3227:
3224:
3218:
3212:
3205:
3202:
3199:
3196:
3178:
3172:
3165:
3162:
3159:
3156:
3138:
3132:
3125:
3122:
3119:
3116:
3098:
3092:
3085:
3077:
3074:
3071:
3068:
3064:
3049:
3046:
3043:
3036:
3028:
3025:
3021:
3017:
3015:
2992:
2986:
2969:
2967:
2959:
2952:
2951:
2950:
2947:2010210120102
2939:1202102012021
2931:0121021201210
2912:
2908:
2877:
2873:
2848:
2845:
2842:
2838:
2809:
2806:
2803:
2798:
2794:
2762:
2758:
2732:
2729:
2726:
2723:
2720:
2709:
2702:
2698:
2690:
2685:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2550:
2549:
2548:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2441:
2434:
2433:
2432:
2430:
2426:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2375:
2371:
2366:
2364:
2362:
2358:
2350:
2348:
2342:
2324:
2320:
2313:
2288:
2285:
2282:
2279:
2273:
2249:
2243:
2237:
2234:
2230:
2225:
2218:
2214:
2210:
2205:
2201:
2198:
2191:
2187:
2183:
2178:
2171:
2167:
2163:
2158:
2151:
2147:
2143:
2138:
2135:
2131:
2127:
2124:
2121:
2092:
2089:
2086:
2072:
2061:
2050:
2040:
2017:
2014:
2008:
2000:
1996:
1969:
1965:
1937:
1934:
1931:
1923:
1919:
1915:
1912:
1890:
1873:
1858:
1855:
1847:
1835:
1831:
1823:
1805:
1802:
1799:
1786:
1770:
1764:
1756:
1739:
1735:
1727:
1709:
1706:
1703:
1675:
1669:
1646:
1643:
1638:
1634:
1609:
1601:
1598:
1595:
1592:
1552:
1548:
1515:
1486:
1482:
1457:
1449:
1446:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1370:
1367:
1364:
1323:
1320:
1317:
1304:
1301:
1288:
1285:
1282:
1273:
1258:
1255:
1252:
1244:
1241:
1237:
1233:
1214:
1211:
1208:
1195:
1187:
1179:
1177:
1078:
1061:
1056:
1052:
1047:
1041:
1037:
1032:
1028:
1023:
1019:
1014:
1008:
1004:
998:
994:
989:
985:
981:
978:
975:
972:
952:
949:
941:
937:
933:
928:
924:
893:
889:
885:
880:
876:
872:
867:
863:
859:
854:
850:
818:is a line of
778:
775:
772:
755:be the entry
740:
737:
734:
730:
675:
632:
625:
622:
619:
616:
613:
610:
607:
606:
602:
599:
596:
593:
590:
587:
580:
579:
575:
572:
569:
566:
563:
560:
557:
556:
552:
549:
546:
543:
540:
537:
530:
529:
519:
508:
502:
500:
498:
494:
478:
458:
455:
452:
449:
446:
421:
417:
407:
401:
395:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
326:
320:
311:
310:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
265:
264:
258:
237:
231:
223:
207:
204:
201:
198:
195:
179:
177:
163:
143:
140:
137:
134:
131:
128:
125:
122:
119:
116:
113:
90:
87:
84:
74:
66:
61:
59:
54:
42:
38:
34:
30:
29:combinatorics
19:
4647:
4612:
4609:Lothaire, M.
4585:
4582:Lothaire, M.
4550:
4512:
4508:
4502:
4485:
4481:
4414:
4410:
4364:
4360:
4351:
4324:
4290:
4286:
4280:
4255:
4251:
4245:
4218:
4214:
4204:
4175:
4171:
4161:
4134:
4130:
4120:
4094:
4088:
4071:
4067:
4045:. Retrieved
4041:
4032:
3983:
3943:
3656:
3648:
3641:
3638:
3630:
3613:
3607:
3604:
3579:
3502:
3499:
3429:
3426:
3415:
3414:square-free
3411:
3380:
3377:
3331:
3285:
3252:
3249:
3203:
3200:
3197:
3194:
3163:
3160:
3157:
3154:
3123:
3120:
3117:
3114:
3083:
3075:
3072:
3069:
3066:
3062:
3034:
3026:
3023:
3019:
2970:
2965:
2963:
2705:
2546:
2378:
2354:
2306:
2073:
2066:
2059:
2048:
2038:
1871:
1833:
1829:
1821:
1757:
1754:
1725:
1302:
1275:
1274:word length
1271:
1242:
1239:
1235:
1185:
1183:
1079:
636:
608:growth rate
558:growth rate
509:
506:
387:
185:
183:
70:
40:
32:
26:
4367:: 277–278.
4195:10995/92700
2941:, and each
2055:letters of
1824:the number
1351:(Note that
626:13.9282035
623:12.9226167
620:11.9160804
617:10.9083279
4681:Categories
4670:1014.11015
4635:1221.68183
4574:1161.68043
4539:References
4456:1505.00019
4389:0079.01101
4047:2019-03-28
3937:of length
3490:increment
3422:of length
3198:k_sf_words
3158:k_sf_words
3118:k_sf_words
3073:k_sf_words
2708:John Leech
2681:(sequence
2030:increment
1531:such that
1345:of length
965:, and for
908:such that
614:9.8989813
611:8.8874856
576:7.8729902
573:6.8541173
570:5.8284661
567:4.7914069
564:3.7325386
561:2.6215080
4529:0304-3975
4417:: 14–25.
4381:126406225
4361:Math. Gaz
4357:Leech, J.
4343:494791187
4307:0196-6774
4272:0020-0190
4237:0304-3975
4178:: 67–72.
4153:0304-3975
4006:≈
3966:≈
3837:∞
3834:→
3529:∞
3526:→
3353:≠
3307:≠
3020:algorithm
2810:∈
2804:∣
2654:−
2633:−
2612:−
2585:−
2570:−
2390:−
2361:Axel Thue
2286:
1997:χ
1966:χ
1920:χ
1887:Σ
1796:Σ
1736:χ
1700:Σ
1635:χ
1606:Σ
1602:∈
1599:136263163
1549:χ
1535:precedes
1512:Σ
1483:χ
1454:Σ
1450:∈
1361:Σ
1256:≥
1236:algorithm
976:≥
794:. A word
479:α
459:1.3017619
453:α
447:1.3017597
418:α
411:Θ
114:ϵ
4611:(2011).
4584:(1997).
4488:(2): 3.
4042:oeis.org
3164:for each
3124:for each
3084:for each
2701:morphism
2367:Examples
2357:alphabet
694:, where
497:automata
439:, where
73:alphabet
47:, where
4490:Bibcode
3027:output:
2933:, each
2927:
2900:
2892:
2865:
2777:
2750:
2748:. Let
2687:in the
2684:A029883
2427:of the
2108:
2076:
1957:update
1874:choose
1662:choose
1339:
1307:
1303:output:
1230:
1198:
342:
313:
4668:
4658:
4633:
4623:
4596:
4572:
4562:
4527:
4429:
4387:
4379:
4341:
4331:
4305:
4270:
4235:
4151:
4109:
4009:0.3905
3969:0.2747
3906:where
3723:where
3430:return
3033:.
2060:return
1832:0
1647:361245
1349:.
1243:input:
156:, and
41:square
4478:(PDF)
4451:arXiv
4377:S2CID
4025:Notes
3621:cbaca
3253:break
3065:True
3063:while
2945:with
2937:with
2929:with
2697:Leech
1834:while
661:from
35:is a
4656:ISBN
4621:ISBN
4594:ISBN
4560:ISBN
4525:ISSN
4427:ISBN
4339:OCLC
4329:ISBN
4303:ISSN
4268:ISSN
4233:ISSN
4149:ISSN
4107:ISBN
3961:3215
3784:and
3618:cbca
3615:bcba
3427:then
3378:then
3250:then
2689:OEIS
2532:0...
2049:then
1856:<
1625:has
1286:>
1238:R2F
456:<
450:<
382:264
379:204
376:144
373:108
37:word
31:, a
4666:Zbl
4631:Zbl
4570:Zbl
4517:doi
4513:380
4419:doi
4415:166
4385:Zbl
4369:doi
4295:doi
4260:doi
4223:doi
4190:hdl
4180:doi
4176:601
4139:doi
4099:doi
4076:doi
4001:653
3998:255
3958:883
3764:in
3644:aba
3494:by
3416:for
3332:and
3070:set
3035:set
2898:in
2699:'s
2283:log
2266:in
2034:by
1878:in
1822:set
1726:set
1691:in
1580:in
1572:in
1539:in
1186:R2F
917:gcd
690:to
603:15
600:14
597:13
594:12
591:11
588:10
370:78
367:60
364:42
361:30
358:18
355:12
307:12
304:11
301:10
164:101
144:010
27:In
4683::
4664:.
4654:.
4629:.
4619:.
4592:.
4568:.
4558:.
4523:.
4511:.
4486:10
4484:.
4480:.
4465:^
4441:^
4425:.
4409:.
4397:^
4383:.
4375:.
4365:41
4363:.
4337:.
4315:^
4301:.
4289:.
4266:.
4256:12
4254:.
4231:.
4219:22
4217:.
4213:.
4188:.
4174:.
4170:.
4147:.
4135:56
4133:.
4129:.
4105:,
4070:.
4056:^
4040:.
4021:.
3981:.
3941:.
3646:.
3626:ab
3610:ab
3594:ab
3541:,
3412:is
3381:if
3286:if
3204:if
3201:do
3195:in
3161:do
3155:in
3121:do
3115:in
3076:to
3067:do
3024:is
3016:.
2691:).
2363:.
2347:.
2114:is
2071:.
2039:if
1872:do
1830:to
1755:to
1660:.)
1473:,
1305:a
1240:is
1077:.
585:)
553:9
550:8
547:7
544:6
541:5
538:4
535:)
352:6
349:3
346:1
298:9
295:8
292:7
289:6
286:5
283:4
280:3
277:2
274:1
271:0
257:.
176:.
138:10
132:01
58:.
56:XX
45:XX
4672:.
4637:.
4604:.
4602:.
4576:.
4531:.
4519::
4496:.
4492::
4459:.
4453::
4435:.
4421::
4391:.
4371::
4345:.
4309:.
4297::
4291:5
4274:.
4262::
4239:.
4225::
4198:.
4192::
4182::
4155:.
4141::
4101::
4082:.
4078::
4072:6
4050:.
3986:a
3946:a
3939:l
3935:w
3919:l
3915:w
3890:|
3884:l
3880:w
3875:|
3867:a
3862:|
3855:l
3851:w
3846:|
3831:l
3816:a
3801:|
3797:w
3793:|
3772:w
3762:a
3746:a
3741:|
3736:w
3732:|
3707:|
3703:w
3699:|
3691:a
3686:|
3681:w
3677:|
3663:w
3659:a
3596:.
3565:)
3562:0
3559:(
3554:w
3550:h
3523:w
3513:h
3509:h
3505:0
3496:1
3492:k
3478:)
3475:2
3472:(
3469:h
3466:,
3463:)
3460:1
3457:(
3454:h
3451:,
3448:)
3445:0
3442:(
3439:h
3424:3
3420:w
3399:)
3396:w
3393:(
3390:h
3365:)
3362:0
3359:(
3356:h
3350:)
3347:2
3344:(
3341:h
3319:)
3316:1
3313:(
3310:h
3304:)
3301:0
3298:(
3295:h
3272:)
3269:1
3266:(
3263:h
3237:)
3234:2
3231:(
3228:h
3225:=
3222:)
3219:1
3216:(
3213:h
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