37:
25:
613:
198:
256:
1204:
1409:
476:. Thus, in either case, the parent is a fraction with a smaller sum of numerator and denominator, so repeated reduction of this type must eventually reach the number 1. As a graph with one outgoing edge per vertex and one root reachable by all other vertices, the Calkin–Wilf tree must indeed be a tree.
1662:
The description here is dual to the original definition by Calkin and Wilf, which begins by defining the child relationship and derives the parent relationship as part of a proof that every rational appears once in the tree. As defined here, every rational appears once by definition, and instead the
1587:
in that both are binary trees with each positive rational number appearing exactly once. Additionally, the top levels of the two trees appear very similar, and in both trees, the same numbers appear at the same levels. One tree can be obtained from the other by performing a
855:
Because the Calkin–Wilf tree contains every positive rational number exactly once, so does this sequence. The denominator of each fraction equals the numerator of the next fraction in the sequence. The Calkin–Wilf sequence can also be generated directly by the formula
1272:
1592:
on the numbers at each level of the trees. Alternatively, the number at a given node of the Calkin–Wilf tree can be converted into the number at the same position in the Stern–Brocot tree, and vice versa, by a process involving the reversal of the
1009:: the number of consecutive 1s starting from the least significant bit, then the number of consecutive 0s starting from the first block of 1s, and so on. The sequence of numbers generated in this way gives the
940:
1277:
1404:{\displaystyle {\begin{aligned}\operatorname {fusc} (2n)&=\operatorname {fusc} (n)\\\operatorname {fusc} (2n+1)&=\operatorname {fusc} (n)+\operatorname {fusc} (n+1),\end{aligned}}}
1195:; however, in the Calkin–Wilf tree the binary numbers are integers (positions in the breadth-first traversal) while in the question mark function they are real numbers between 0 and 1.
1706:, then the fractions on the level just below the top of the tree, reading from left to right, then the fractions on the next level down, reading from left to right, etc."
596:: its inorder does not coincide with the sorted order of its vertices. However, it is closely related to a different binary search tree on the same set of vertices, the
1958:
1601:: the left-to-right traversal order of the tree is the same as the numerical order of the numbers in it. This property is not true of the Calkin–Wilf tree.
1516:
in which each power occurs at most twice. This can be seen from the recurrence defining fusc: the expressions as a sum of powers of two for an even number
1234:
1457:. Thus, the diatomic sequence forms both the sequence of numerators and the sequence of denominators of the numbers in the Calkin–Wilf sequence.
293:
occurs as a vertex and has one outgoing edge to another vertex, its parent, except for the root of the tree, the number 1, which has no parent.
1881:
1836:
1192:
862:
1925:
479:
The children of any vertex in the Calkin–Wilf tree may be computed by inverting the formula for the parents of a vertex. Each vertex
2068:
2063:
2009:
1597:
representations of these numbers. However, in other ways, they have different properties: for instance, the Stern–Brocot tree is a
2238:
1776:
and to uncited work of Lind. However, Carlitz's paper describes a more restricted class of sums of powers of two, counted by
1916:
230:, since they drew some ideas from a 1973 paper by George N. Raney. Stern's diatomic series was formulated much earlier by
982:
1191:
A similar conversion between run-length-encoded binary numbers and continued fractions can also be used to evaluate
2233:
296:
The parent of any rational number can be determined after placing the number into simplest terms, as a fraction
1589:
601:
319:
1690:: "a list of all positive rational numbers, each appearing once and only once, can be made by writing down
1529:) or two 1s (in which case the rest of the expression is formed by doubling each term in an expression for
36:
1711:
77:
1584:
597:
235:
2101:
1002:
62:
2139:
1827:
1822:
1646:
1469:
1242:
231:
174:
1523:
either have no 1s in them (in which case they are formed by doubling each term in an expression for
624:
is the sequence of rational numbers generated by a breadth-first traversal of the Calkin–Wilf tree,
2207:
2148:
1264:
998:
58:
2128:
2087:
2048:
1989:
1937:
1736:
1598:
1594:
1567:
6 = 4 + 2 = 4 + 1 + 1 = 2 + 2 + 1 + 1
1172:
1133:
1010:
593:
76:. The tree is rooted at the number 1, and any rational number expressed in simplest terms as the
2189:
2170:
2005:
1832:
1571:
has three representations as a sum of powers of two with at most two copies of each power, so
2143:
1263:, named according to the obfuscating appearance of the sequence of values and defined by the
592:
As each vertex has two children, the Calkin–Wilf tree is a binary tree. However, it is not a
2118:
2110:
2077:
2040:
1967:
1953:
1929:
1890:
1858:
1221:
243:
206:
166:
1981:
1872:
600:: the vertices at each level of the two trees coincide, and are related to each other by a
2193:
2001:
1977:
1949:
1868:
1642:
239:
202:
162:
73:
70:
24:
1908:
616:
The Calkin–Wilf sequence, depicted as the red spiral tracing through the Calkin–Wilf tree
238:. Even earlier, a similar tree (including only the fractions between 0 and 1) appears in
2020:
2016:
2059:
1994:
268:
1895:
1431:
th rational number in a breadth-first traversal of the Calkin–Wilf tree is the number
2227:
2174:
2132:
1818:
153:. Every positive rational number appears exactly once in the tree. It is named after
50:
2091:
1972:
1536:), so the number of representations is the sum of the number of representations for
197:
2217:
1904:
1513:
1207:
223:
219:
158:
612:
215:
154:
1879:
Berstel, Jean; de Luca, Aldo (1997), "Sturmian words, Lyndon words and trees",
255:
2211:
2123:
2082:
1863:
1176:
1175:
is . But to use this method, the length of the continued fraction must be an
2198:
2179:
1549:, matching the recurrence. Similarly, each representation for an odd number
1203:
234:, a 19th-century German mathematician who also invented the closely related
66:
1846:
2114:
2099:
Raney, George N. (1973), "On continued fractions and finite automata",
2052:
2028:
1941:
1179:. So should be replaced by the equivalent continued fraction . Hence
260:
2044:
2027:
Knuth, Donald E.; Rupert, C.P.; Smith, Alex; Stong, Richard (2003),
1933:
935:{\displaystyle q_{i+1}={\frac {1}{2\lfloor q_{i}\rfloor -q_{i}+1}}}
1202:
611:
254:
196:
181:. Its sequence of numerators (or, offset by one, denominators) is
2023:, pp. 230–232, reprints of notes originally written in 1976.
1671:
1669:
1227:
0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, … (sequence
1663:
fact that the resulting structure is a tree requires a proof.
2021:
EWD 578: More about the function "fusc" (A sequel to EWD570)
1831:(3rd ed.), Berlin; New York: Springer, pp. 94–97,
1091:
In the other direction, using the continued fraction of any
1563:
and adding 1, again matching the recurrence. For instance,
1229:
1100:
as the run-length encoding of a binary number gives back
2066:(2006), "Functional pearl: Enumerating the rationals",
1714:
techniques for performing this breadth first traversal.
1996:
Selected
Writings on Computing: A Personal Perspective
1845:
Bates, Bruce; Bunder, Martin; Tognetti, Keith (2010),
222:, who considered it in a 2000 paper. In a 1997 paper,
1723:
1275:
865:
1801:
1993:
1403:
934:
1707:
1675:
1952:(1964), "A problem in partitions related to the
1847:"Linking the Calkin-Wilf and Stern-Brocot trees"
1506:, and also counts the number of ways of writing
2149:Journal für die reine und angewandte Mathematik
1617:
1959:Bulletin of the American Mathematical Society
563:has one child whose value is greater than 1,
8:
2029:"Recounting the Rationals, Continued: 10906"
907:
894:
1557:is formed by doubling a representation for
1748:
1687:
501:has one child whose value is less than 1,
226:and Aldo de Luca called the same tree the
2122:
2081:
1971:
1894:
1862:
1276:
1274:
960:th number in the sequence, starting from
917:
901:
885:
870:
864:
267:The Calkin–Wilf tree may be defined as a
2144:"Ueber eine zahlentheoretische Funktion"
2017:EWD 570: An exercise for Dr.R.M.Burstall
177:of the Calkin–Wilf tree is known as the
42:How values are derived from their parent
1773:
1760:
1610:
271:in which each positive rational number
161:, but appears in other works including
1251:th value in the sequence is the value
173:The sequence of rational numbers in a
1629:
259:The Calkin–Wilf tree, drawn using an
7:
1772:The OEIS entry credits this fact to
1726:credit this formula to Moshe Newman.
214:The Calkin–Wilf tree is named after
101:has as its two children the numbers
1926:Mathematical Association of America
1802:Bates, Bunder & Tognetti (2010)
1735:The fusc name was given in 1976 by
1583:The Calkin–Wilf tree resembles the
1062:: The continued fraction is hence
1030:: The continued fraction is hence
1193:Minkowski's question mark function
14:
2069:Journal of Functional Programming
2033:The American Mathematical Monthly
1851:European Journal of Combinatorics
1708:Gibbons, Lester & Bird (2006)
1676:Gibbons, Lester & Bird (2006)
1198:
988:It's also possible to calculate
35:
23:
1973:10.1090/S0002-9904-1964-11118-6
16:Binary tree of rational numbers
1391:
1379:
1367:
1361:
1345:
1330:
1317:
1311:
1295:
1286:
1:
2213:Fractions on a Binary Tree II
1917:American Mathematical Monthly
1896:10.1016/S0304-3975(96)00101-6
1579:Relation to Stern–Brocot tree
185:, and can be computed by the
1882:Theoretical Computer Science
1618:Berstel & de Luca (1997)
1651:, vol. III, p. 27
2255:
1909:"Recounting the rationals"
2194:"Stern's Diatomic Series"
2083:10.1017/S0956796806005880
1864:10.1016/j.ejc.2010.04.002
1218:Stern's diatomic sequence
1199:Stern's diatomic sequence
541:. Similarly, each vertex
2019:, pp. 215–216, and
1749:Calkin & Wilf (2000)
1739:; see EWD570 and EWD578.
1688:Calkin & Wilf (2000)
1590:bit-reversal permutation
602:bit-reversal permutation
251:Definition and structure
2239:Trees (data structures)
608:Breadth first traversal
320:greatest common divisor
183:Stern's diatomic series
175:breadth-first traversal
1712:functional programming
1405:
1214:
1058:i = 1990 = 11111000110
1026:i = 1081 = 10000111001
936:
617:
264:
211:
2102:Mathematische Annalen
1470:binomial coefficients
1468:is the number of odd
1406:
1206:
1003:binary representation
937:
615:
258:
200:
2208:Bogomolny, Alexander
1828:Proofs from THE BOOK
1414:with the base cases
1273:
1265:recurrence relations
1243:zero-based numbering
863:
622:Calkin–Wilf sequence
527:, because of course
232:Moritz Abraham Stern
179:Calkin–Wilf sequence
30:The Calkin–Wilf tree
1990:Dijkstra, Edsger W.
1724:Knuth et al. (2003)
999:run-length encoding
2190:Weisstein, Eric W.
2175:"Calkin–Wilf Tree"
2171:Weisstein, Eric W.
2124:10338.dmlcz/128216
2115:10.1007/BF01355980
1823:Ziegler, Günter M.
1737:Edsger W. Dijkstra
1710:discuss efficient
1599:binary search tree
1595:continued fraction
1401:
1399:
1215:
1173:continued fraction
1134:continued fraction
1013:representation of
1011:continued fraction
997:directly from the
932:
618:
594:binary search tree
265:
212:
2234:Integer sequences
2062:; Lester, David;
1838:978-3-540-40460-6
1585:Stern–Brocot tree
1104:itself. Example:
930:
598:Stern–Brocot tree
236:Stern–Brocot tree
2246:
2220:
2203:
2202:
2184:
2183:
2157:
2140:Stern, Moritz A.
2135:
2126:
2094:
2085:
2055:
2014:
1999:
1984:
1975:
1954:Stirling numbers
1944:
1913:
1899:
1898:
1889:(1–2): 171–203,
1875:
1866:
1857:(7): 1637–1661,
1841:
1805:
1799:
1793:
1791:
1783:
1770:
1764:
1758:
1752:
1746:
1740:
1733:
1727:
1721:
1715:
1705:
1703:
1702:
1699:
1696:
1685:
1679:
1673:
1664:
1660:
1654:
1652:
1648:Harmonices Mundi
1639:
1633:
1627:
1621:
1615:
1574:
1562:
1556:
1548:
1541:
1535:
1528:
1522:
1511:
1505:
1494:
1488:
1487:
1467:
1456:
1455:
1453:
1452:
1445:
1442:
1430:
1421:
1417:
1410:
1408:
1407:
1402:
1400:
1258:
1250:
1232:
1222:integer sequence
1213:
1182:
1170:
1169:
1167:
1166:
1163:
1160:
1139:
1131:
1130:
1128:
1127:
1124:
1121:
1103:
1099:
1086:
1085:
1083:
1082:
1079:
1076:
1054:
1053:
1051:
1050:
1047:
1044:
1021:
1008:
996:
980:
969:
959:
953:
941:
939:
938:
933:
931:
929:
922:
921:
906:
905:
886:
881:
880:
850:
848:
847:
844:
841:
834:
832:
831:
828:
825:
818:
816:
815:
812:
809:
802:
800:
799:
796:
793:
786:
784:
783:
780:
777:
770:
768:
767:
764:
761:
754:
752:
751:
748:
745:
738:
736:
735:
732:
729:
722:
720:
719:
716:
713:
706:
704:
703:
700:
697:
690:
688:
687:
684:
681:
674:
672:
671:
668:
665:
658:
656:
655:
652:
649:
642:
640:
639:
636:
633:
588:
587:
585:
584:
579:
576:
562:
561:
559:
558:
553:
550:
540:
526:
525:
523:
522:
513:
510:
500:
499:
497:
496:
491:
488:
475:
474:
472:
471:
466:
463:
449:
448:
446:
445:
440:
437:
428:, the parent of
427:
425:
423:
422:
417:
414:
404:
403:
401:
400:
391:
388:
378:
377:
375:
374:
369:
366:
357:, the parent of
356:
354:
352:
351:
346:
343:
333:
327:
317:
316:
314:
313:
308:
305:
292:
291:
289:
288:
283:
280:
244:Harmonices Mundi
207:Harmonices Mundi
167:Harmonices Mundi
152:
151:
149:
148:
143:
140:
126:
125:
123:
122:
113:
110:
100:
99:
97:
96:
91:
88:
74:rational numbers
55:Calkin–Wilf tree
39:
27:
2254:
2253:
2249:
2248:
2247:
2245:
2244:
2243:
2224:
2223:
2206:
2188:
2187:
2169:
2168:
2165:
2138:
2098:
2060:Gibbons, Jeremy
2058:
2045:10.2307/3647762
2026:
2012:
2002:Springer-Verlag
1988:
1948:
1934:10.2307/2589182
1911:
1902:
1878:
1844:
1839:
1817:
1814:
1809:
1808:
1800:
1796:
1785:
1777:
1771:
1767:
1759:
1755:
1747:
1743:
1734:
1730:
1722:
1718:
1700:
1697:
1694:
1693:
1691:
1686:
1682:
1674:
1667:
1661:
1657:
1641:
1640:
1636:
1628:
1624:
1616:
1612:
1607:
1581:
1573:fusc(6 + 1) = 3
1572:
1558:
1550:
1543:
1537:
1530:
1524:
1517:
1507:
1496:
1493:
1492:
1486:
1481:
1480:
1479:
1478:
1477:
1473:
1461:
1446:
1443:
1436:
1435:
1433:
1432:
1426:
1419:
1415:
1398:
1397:
1348:
1321:
1320:
1298:
1271:
1270:
1252:
1246:
1228:
1211:
1201:
1186:
1180:
1164:
1161:
1158:
1157:
1155:
1152:
1147:
1143:
1137:
1125:
1122:
1119:
1118:
1116:
1113:
1108:
1101:
1097:
1092:
1080:
1077:
1074:
1073:
1071:
1069:
1063:
1061:
1048:
1045:
1042:
1041:
1039:
1037:
1031:
1029:
1019:
1014:
1006:
994:
989:
981:represents the
977:
971:
967:
961:
955:
951:
946:
913:
897:
890:
866:
861:
860:
845:
842:
839:
838:
836:
829:
826:
823:
822:
820:
813:
810:
807:
806:
804:
797:
794:
791:
790:
788:
781:
778:
775:
774:
772:
765:
762:
759:
758:
756:
749:
746:
743:
742:
740:
733:
730:
727:
726:
724:
717:
714:
711:
710:
708:
701:
698:
695:
694:
692:
685:
682:
679:
678:
676:
669:
666:
663:
662:
660:
653:
650:
647:
646:
644:
637:
634:
631:
630:
628:
610:
580:
577:
568:
567:
565:
564:
554:
551:
546:
545:
543:
542:
528:
514:
511:
506:
505:
503:
502:
492:
489:
484:
483:
481:
480:
467:
464:
455:
454:
452:
451:
441:
438:
433:
432:
430:
429:
418:
415:
410:
409:
407:
406:
392:
389:
384:
383:
381:
380:
370:
367:
362:
361:
359:
358:
347:
344:
339:
338:
336:
335:
329:
323:
309:
306:
301:
300:
298:
297:
284:
281:
276:
275:
273:
272:
253:
195:
144:
141:
132:
131:
129:
128:
114:
111:
106:
105:
103:
102:
92:
89:
84:
83:
81:
80:
47:
46:
45:
44:
43:
40:
32:
31:
28:
17:
12:
11:
5:
2252:
2250:
2242:
2241:
2236:
2226:
2225:
2222:
2221:
2204:
2185:
2164:
2163:External links
2161:
2160:
2159:
2136:
2109:(4): 265–283,
2096:
2076:(3): 281–291,
2056:
2039:(7): 642–643,
2024:
2010:
1986:
1966:(2): 275–278,
1946:
1903:Calkin, Neil;
1900:
1876:
1842:
1837:
1819:Aigner, Martin
1813:
1810:
1807:
1806:
1794:
1784:instead of by
1774:Carlitz (1964)
1765:
1761:Carlitz (1964)
1753:
1741:
1728:
1716:
1680:
1665:
1655:
1634:
1622:
1609:
1608:
1606:
1603:
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1568:
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1347:
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1341:
1338:
1335:
1332:
1329:
1326:
1323:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1299:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1278:
1239:
1238:
1212:fusc(0...4096)
1200:
1197:
1189:
1188:
1184:
1150:
1145:
1141:
1111:
1095:
1089:
1088:
1067:
1059:
1056:
1035:
1027:
1017:
992:
975:
965:
949:
943:
942:
928:
925:
920:
916:
912:
909:
904:
900:
896:
893:
889:
884:
879:
876:
873:
869:
853:
852:
609:
606:
269:directed graph
252:
249:
201:The tree from
194:
191:
41:
34:
33:
29:
22:
21:
20:
19:
18:
15:
13:
10:
9:
6:
4:
3:
2:
2251:
2240:
2237:
2235:
2232:
2231:
2229:
2219:
2215:
2214:
2209:
2205:
2201:
2200:
2195:
2191:
2186:
2182:
2181:
2176:
2172:
2167:
2166:
2162:
2155:
2151:
2150:
2145:
2141:
2137:
2134:
2130:
2125:
2120:
2116:
2112:
2108:
2104:
2103:
2097:
2093:
2089:
2084:
2079:
2075:
2071:
2070:
2065:
2064:Bird, Richard
2061:
2057:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2025:
2022:
2018:
2013:
2011:0-387-90652-5
2007:
2003:
1998:
1997:
1991:
1987:
1983:
1979:
1974:
1969:
1965:
1961:
1960:
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1923:
1919:
1918:
1910:
1906:
1905:Wilf, Herbert
1901:
1897:
1892:
1888:
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1883:
1877:
1874:
1870:
1865:
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1856:
1852:
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1729:
1725:
1720:
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1713:
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1689:
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1677:
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1666:
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1644:
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1596:
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1576:
1566:
1565:
1564:
1561:
1554:
1546:
1540:
1533:
1527:
1521:
1515:
1514:powers of two
1510:
1504:
1500:
1485:
1471:
1465:
1460:The function
1458:
1450:
1440:
1429:
1423:
1394:
1388:
1385:
1382:
1376:
1373:
1370:
1364:
1358:
1355:
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1342:
1339:
1336:
1333:
1327:
1324:
1314:
1308:
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1302:
1300:
1292:
1289:
1283:
1280:
1269:
1268:
1267:
1266:
1262:
1261:fusc function
1256:
1249:
1244:
1236:
1231:
1226:
1225:
1224:
1223:
1219:
1209:
1205:
1196:
1194:
1178:
1174:
1153:
1146:
1135:
1114:
1107:
1106:
1105:
1098:
1066:
1057:
1034:
1025:
1024:
1023:
1020:
1012:
1004:
1000:
995:
986:
984:
983:integral part
978:
964:
958:
952:
926:
923:
918:
914:
910:
902:
898:
891:
887:
882:
877:
874:
871:
867:
859:
858:
857:
627:
626:
625:
623:
614:
607:
605:
603:
599:
595:
590:
583:
575:
571:
557:
549:
539:
535:
531:
521:
517:
509:
495:
487:
477:
470:
462:
458:
444:
436:
421:
413:
399:
395:
387:
373:
365:
350:
342:
332:
326:
321:
312:
304:
294:
287:
279:
270:
262:
257:
250:
248:
246:
245:
241:
237:
233:
229:
225:
221:
217:
209:
208:
204:
199:
192:
190:
188:
187:fusc function
184:
180:
176:
171:
169:
168:
164:
160:
156:
147:
139:
135:
121:
117:
109:
95:
87:
79:
75:
72:
68:
64:
61:in which the
60:
56:
52:
51:number theory
38:
26:
2218:Cut-the-knot
2212:
2197:
2178:
2153:
2147:
2106:
2100:
2073:
2067:
2036:
2032:
1995:
1963:
1957:
1921:
1915:
1886:
1880:
1854:
1850:
1826:
1797:
1787:
1779:
1768:
1756:
1751:, Theorem 1.
1744:
1731:
1719:
1683:
1658:
1647:
1637:
1630:Raney (1973)
1625:
1620:, Section 6.
1613:
1582:
1570:
1559:
1552:
1544:
1538:
1531:
1525:
1519:
1512:as a sum of
1508:
1502:
1498:
1483:
1472:of the form
1463:
1459:
1448:
1438:
1427:
1424:
1413:
1260:
1254:
1247:
1240:
1217:
1216:
1190:
1148:
1109:
1093:
1090:
1064:
1032:
1015:
990:
987:
973:
962:
956:
954:denotes the
947:
944:
854:
621:
619:
591:
581:
573:
569:
555:
547:
537:
533:
529:
519:
515:
507:
493:
485:
478:
468:
460:
456:
442:
434:
419:
411:
397:
393:
385:
371:
363:
348:
340:
330:
324:
310:
302:
295:
285:
277:
266:
242:
227:
224:Jean Berstel
220:Herbert Wilf
213:
205:
186:
182:
178:
172:
165:
159:Herbert Wilf
145:
137:
133:
119:
115:
107:
93:
85:
54:
48:
1950:Carlitz, L.
1928:: 360–363,
1420:fusc(1) = 1
1416:fusc(0) = 0
1208:Scatterplot
216:Neil Calkin
155:Neil Calkin
65:correspond
2228:Categories
1812:References
1643:Kepler, J.
1177:odd number
1136:is hence
1022:.Example:
318:for which
228:Raney tree
67:one-to-one
2199:MathWorld
2180:MathWorld
2156:: 193–220
2133:120933574
1377:
1359:
1328:
1309:
1284:
911:−
908:⌋
895:⌊
334:is 1. If
2142:(1858),
2092:14237968
1992:(1982),
1907:(2000),
1825:(2004),
1645:(1619),
1542:and for
247:(1619).
240:Kepler's
203:Kepler's
163:Kepler's
78:fraction
71:positive
63:vertices
2053:3647762
1982:0157907
1942:2589182
1873:2673006
1704:
1692:
1454:
1434:
1259:of the
1233:in the
1230:A002487
1220:is the
1168:
1156:
1129:
1117:
1084:
1072:
1052:
1040:
1001:of the
849:
837:
833:
821:
817:
805:
801:
789:
785:
773:
769:
757:
753:
741:
737:
725:
721:
709:
705:
693:
689:
677:
673:
661:
657:
645:
641:
629:
586:
566:
560:
544:
524:
504:
498:
482:
473:
453:
447:
431:
424:
408:
402:
382:
376:
360:
353:
337:
315:
299:
290:
274:
263:layout.
193:History
150:
130:
124:
104:
98:
82:
69:to the
2131:
2090:
2051:
2008:
1980:
1940:
1871:
1835:
1245:, the
1241:Using
1183:= 1001
1171:: The
1140:= 1110
1132:: The
970:, and
945:where
426:> 1
355:< 1
261:H tree
210:(1619)
53:, the
2129:S2CID
2088:S2CID
2049:JSTOR
1938:JSTOR
1924:(4),
1912:(PDF)
1786:fusc(
1778:fusc(
1605:Notes
1501:<
1497:0 ≤ 2
1462:fusc(
1447:fusc(
1437:fusc(
1253:fusc(
1144:= 14.
536:>
405:; if
57:is a
2006:ISBN
1833:ISBN
1790:+ 1)
1466:+ 1)
1451:+ 1)
1425:The
1418:and
1374:fusc
1356:fusc
1325:fusc
1306:fusc
1281:fusc
1235:OEIS
1187:= 9.
1068:1990
1036:1081
851:, ….
620:The
328:and
218:and
157:and
127:and
59:tree
2119:hdl
2111:doi
2107:206
2078:doi
2041:doi
2037:110
1968:doi
1956:",
1930:doi
1922:107
1891:doi
1887:178
1859:doi
1555:+ 1
1547:− 1
1534:− 1
1210:of
1005:of
968:= 1
450:is
379:is
322:of
49:In
2230::
2216:,
2210:,
2196:,
2192:,
2177:,
2173:,
2154:55
2152:,
2146:,
2127:,
2117:,
2105:,
2086:,
2074:16
2072:,
2047:,
2035:,
2031:,
2015:.
2004:,
2000:,
1978:MR
1976:,
1964:70
1962:,
1936:,
1920:,
1914:,
1885:,
1869:MR
1867:,
1855:31
1853:,
1849:,
1821:;
1668:^
1575:.
1495:,
1422:.
1237:).
1154:=
1115:=
1081:53
1075:37
1070:=
1049:37
1043:53
1038:=
985:.
972:⌊
835:,
819:,
803:,
787:,
771:,
755:,
739:,
723:,
707:,
691:,
675:,
659:,
643:,
604:.
589:.
572:+
532:+
518:+
459:−
396:−
189:.
170:.
136:+
118:+
2158:.
2121::
2113::
2095:.
2080::
2043::
1985:.
1970::
1945:.
1932::
1893::
1861::
1804:.
1792:.
1788:n
1782:)
1780:n
1763:.
1701:1
1698:/
1695:1
1678:.
1653:.
1632:.
1560:n
1553:n
1551:2
1545:n
1539:n
1532:n
1526:n
1520:n
1518:2
1509:n
1503:n
1499:r
1491:)
1484:r
1476:(
1464:n
1449:n
1444:/
1441:)
1439:n
1428:n
1395:,
1392:)
1389:1
1386:+
1383:n
1380:(
1371:+
1368:)
1365:n
1362:(
1353:=
1346:)
1343:1
1340:+
1337:n
1334:2
1331:(
1318:)
1315:n
1312:(
1303:=
1296:)
1293:n
1290:2
1287:(
1257:)
1255:n
1248:n
1185:2
1181:i
1165:3
1162:/
1159:4
1151:i
1149:q
1142:2
1138:i
1126:4
1123:/
1120:3
1112:i
1110:q
1102:i
1096:i
1094:q
1087:.
1078:/
1065:q
1060:2
1055:.
1046:/
1033:q
1028:2
1018:i
1016:q
1007:i
993:i
991:q
979:⌋
976:i
974:q
966:1
963:q
957:i
950:i
948:q
927:1
924:+
919:i
915:q
903:i
899:q
892:2
888:1
883:=
878:1
875:+
872:i
868:q
846:4
843:/
840:3
830:3
827:/
824:5
814:5
811:/
808:2
798:2
795:/
792:5
782:5
779:/
776:3
766:3
763:/
760:4
750:4
747:/
744:1
734:1
731:/
728:3
718:3
715:/
712:2
702:2
699:/
696:3
686:3
683:/
680:1
670:1
667:/
664:2
654:2
651:/
648:1
638:1
635:/
632:1
582:b
578:/
574:b
570:a
556:b
552:/
548:a
538:a
534:b
530:a
520:b
516:a
512:/
508:a
494:b
490:/
486:a
469:b
465:/
461:b
457:a
443:b
439:/
435:a
420:b
416:/
412:a
398:a
394:b
390:/
386:a
372:b
368:/
364:a
349:b
345:/
341:a
331:b
325:a
311:b
307:/
303:a
286:b
282:/
278:a
146:b
142:/
138:b
134:a
120:b
116:a
112:/
108:a
94:b
90:/
86:a
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