48:
36:
624:
209:
267:
1215:
1420:
487:. 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.
1673:
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
1598:
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
866:
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
1283:
1603:
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
1020:: 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
951:
1288:
1415:{\displaystyle {\begin{aligned}\operatorname {fusc} (2n)&=\operatorname {fusc} (n)\\\operatorname {fusc} (2n+1)&=\operatorname {fusc} (n)+\operatorname {fusc} (n+1),\end{aligned}}}
1206:; 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.
1717:, 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."
607:: 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
1969:
1612:: 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.
1527:
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
1245:
1468:. Thus, the diatomic sequence forms both the sequence of numerators and the sequence of denominators of the numbers in the Calkin–Wilf sequence.
304:
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.
1892:
1847:
1203:
873:
1936:
490:
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
2079:
2074:
2020:
1608:
representations of these numbers. However, in other ways, they have different properties: for instance, the Stern–Brocot tree is a
2249:
1787:
and to uncited work of Lind. However, Carlitz's paper describes a more restricted class of sums of powers of two, counted by
1927:
241:, since they drew some ideas from a 1973 paper by George N. Raney. Stern's diatomic series was formulated much earlier by
993:
1202:
A similar conversion between run-length-encoded binary numbers and continued fractions can also be used to evaluate
2244:
307:
The parent of any rational number can be determined after placing the number into simplest terms, as a fraction
1600:
612:
330:
1701:: "a list of all positive rational numbers, each appearing once and only once, can be made by writing down
1540:) or two 1s (in which case the rest of the expression is formed by doubling each term in an expression for
47:
1722:
88:
1595:
608:
246:
2112:
1013:
73:
2150:
1838:
1833:
1657:
1480:
1253:
242:
185:
1534:
either have no 1s in them (in which case they are formed by doubling each term in an expression for
635:
is the sequence of rational numbers generated by a breadth-first traversal of the Calkin–Wilf tree,
2218:
2159:
1275:
1009:
69:
2139:
2098:
2059:
2000:
1948:
1747:
1609:
1605:
1578:
6 = 4 + 2 = 4 + 1 + 1 = 2 + 2 + 1 + 1
1183:
1144:
1021:
604:
87:. The tree is rooted at the number 1, and any rational number expressed in simplest terms as the
2200:
2181:
2016:
1843:
1582:
has three representations as a sum of powers of two with at most two copies of each power, so
17:
2154:
1274:, named according to the obfuscating appearance of the sequence of values and defined by the
603:
As each vertex has two children, the Calkin–Wilf tree is a binary tree. However, it is not a
2129:
2121:
2088:
2051:
1978:
1964:
1940:
1901:
1869:
1232:
254:
217:
177:
1992:
1883:
611:: the vertices at each level of the two trees coincide, and are related to each other by a
2204:
2012:
1988:
1960:
1879:
1653:
250:
213:
173:
84:
81:
35:
1919:
627:
The Calkin–Wilf sequence, depicted as the red spiral tracing through the Calkin–Wilf tree
249:. Even earlier, a similar tree (including only the fractions between 0 and 1) appears in
2031:
2027:
2070:
2005:
279:
1906:
1442:
th rational number in a breadth-first traversal of the Calkin–Wilf tree is the number
2238:
2185:
2143:
1829:
164:. Every positive rational number appears exactly once in the tree. It is named after
61:
2102:
1983:
1547:), so the number of representations is the sum of the number of representations for
208:
2228:
1915:
1524:
1218:
234:
230:
169:
623:
226:
165:
1890:
Berstel, Jean; de Luca, Aldo (1997), "Sturmian words, Lyndon words and trees",
266:
2222:
2134:
2093:
1874:
1187:
1186:
is . But to use this method, the length of the continued fraction must be an
2209:
2190:
1560:, matching the recurrence. Similarly, each representation for an odd number
1214:
245:, a 19th-century German mathematician who also invented the closely related
77:
1857:
2125:
2110:
Raney, George N. (1973), "On continued fractions and finite automata",
2063:
2039:
1952:
1190:. So should be replaced by the equivalent continued fraction . Hence
271:
2055:
2038:
Knuth, Donald E.; Rupert, C.P.; Smith, Alex; Stong, Richard (2003),
1944:
946:{\displaystyle q_{i+1}={\frac {1}{2\lfloor q_{i}\rfloor -q_{i}+1}}}
1213:
622:
265:
207:
192:. Its sequence of numerators (or, offset by one, denominators) is
2034:, pp. 230–232, reprints of notes originally written in 1976.
1682:
1680:
1238:
0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, … (sequence
1674:
fact that the resulting structure is a tree requires a proof.
2032:
EWD 578: More about the function "fusc" (A sequel to EWD570)
1842:(3rd ed.), Berlin; New York: Springer, pp. 94–97,
1102:
In the other direction, using the continued fraction of any
1574:
and adding 1, again matching the recurrence. For instance,
1240:
1111:
as the run-length encoding of a binary number gives back
2077:(2006), "Functional pearl: Enumerating the rationals",
1725:
techniques for performing this breadth first traversal.
2007:
Selected
Writings on Computing: A Personal Perspective
1856:
Bates, Bruce; Bunder, Martin; Tognetti, Keith (2010),
233:, who considered it in a 2000 paper. In a 1997 paper,
1734:
1286:
876:
1812:
2004:
1414:
945:
1718:
1686:
1963:(1964), "A problem in partitions related to the
1858:"Linking the Calkin-Wilf and Stern-Brocot trees"
1517:, and also counts the number of ways of writing
2160:Journal für die reine und angewandte Mathematik
1628:
1970:Bulletin of the American Mathematical Society
574:has one child whose value is greater than 1,
8:
2040:"Recounting the Rationals, Continued: 10906"
918:
905:
1568:is formed by doubling a representation for
1759:
1698:
512:has one child whose value is less than 1,
237:and Aldo de Luca called the same tree the
2133:
2092:
1982:
1905:
1873:
1287:
1285:
971:th number in the sequence, starting from
928:
912:
896:
881:
875:
278:The Calkin–Wilf tree may be defined as a
2155:"Ueber eine zahlentheoretische Funktion"
2028:EWD 570: An exercise for Dr.R.M.Burstall
188:of the Calkin–Wilf tree is known as the
53:How values are derived from their parent
1784:
1771:
1621:
282:in which each positive rational number
172:, but appears in other works including
1262:th value in the sequence is the value
184:The sequence of rational numbers in a
1640:
270:The Calkin–Wilf tree, drawn using an
7:
1783:The OEIS entry credits this fact to
1737:credit this formula to Moshe Newman.
225:The Calkin–Wilf tree is named after
112:has as its two children the numbers
1937:Mathematical Association of America
1813:Bates, Bunder & Tognetti (2010)
1746:The fusc name was given in 1976 by
1594:The Calkin–Wilf tree resembles the
1073:: The continued fraction is hence
1041:: The continued fraction is hence
1204:Minkowski's question mark function
25:
2080:Journal of Functional Programming
2044:The American Mathematical Monthly
1862:European Journal of Combinatorics
1719:Gibbons, Lester & Bird (2006)
1687:Gibbons, Lester & Bird (2006)
1209:
999:It's also possible to calculate
46:
34:
1984:10.1090/S0002-9904-1964-11118-6
27:Binary tree of rational numbers
1402:
1390:
1378:
1372:
1356:
1341:
1328:
1322:
1306:
1297:
18:Stern's diatomic sequence
1:
2224:Fractions on a Binary Tree II
1928:American Mathematical Monthly
1907:10.1016/S0304-3975(96)00101-6
1590:Relation to Stern–Brocot tree
196:, and can be computed by the
1893:Theoretical Computer Science
1629:Berstel & de Luca (1997)
1662:, vol. III, p. 27
2266:
1920:"Recounting the rationals"
2205:"Stern's Diatomic Series"
2094:10.1017/S0956796806005880
1875:10.1016/j.ejc.2010.04.002
1229:Stern's diatomic sequence
1210:Stern's diatomic sequence
552:. Similarly, each vertex
2030:, pp. 215–216, and
1760:Calkin & Wilf (2000)
1750:; see EWD570 and EWD578.
1699:Calkin & Wilf (2000)
1601:bit-reversal permutation
613:bit-reversal permutation
262:Definition and structure
2250:Trees (data structures)
619:Breadth first traversal
331:greatest common divisor
194:Stern's diatomic series
186:breadth-first traversal
1723:functional programming
1416:
1225:
1069:i = 1990 = 11111000110
1037:i = 1081 = 10000111001
947:
628:
275:
222:
2113:Mathematische Annalen
1481:binomial coefficients
1479:is the number of odd
1417:
1217:
1014:binary representation
948:
626:
269:
211:
2219:Bogomolny, Alexander
1839:Proofs from THE BOOK
1425:with the base cases
1284:
1276:recurrence relations
1254:zero-based numbering
874:
633:Calkin–Wilf sequence
538:, because of course
243:Moritz Abraham Stern
190:Calkin–Wilf sequence
41:The Calkin–Wilf tree
2001:Dijkstra, Edsger W.
1735:Knuth et al. (2003)
1010:run-length encoding
2201:Weisstein, Eric W.
2186:"Calkin–Wilf Tree"
2182:Weisstein, Eric W.
2135:10338.dmlcz/128216
2126:10.1007/BF01355980
1834:Ziegler, Günter M.
1748:Edsger W. Dijkstra
1721:discuss efficient
1610:binary search tree
1606:continued fraction
1412:
1410:
1226:
1184:continued fraction
1145:continued fraction
1024:representation of
1022:continued fraction
1008:directly from the
943:
629:
605:binary search tree
276:
223:
2245:Integer sequences
2073:; Lester, David;
1849:978-3-540-40460-6
1596:Stern–Brocot tree
1115:itself. Example:
941:
609:Stern–Brocot tree
247:Stern–Brocot tree
16:(Redirected from
2257:
2231:
2214:
2213:
2195:
2194:
2168:
2151:Stern, Moritz A.
2146:
2137:
2105:
2096:
2066:
2025:
2010:
1995:
1986:
1965:Stirling numbers
1955:
1924:
1910:
1909:
1900:(1–2): 171–203,
1886:
1877:
1868:(7): 1637–1661,
1852:
1816:
1810:
1804:
1802:
1794:
1781:
1775:
1769:
1763:
1757:
1751:
1744:
1738:
1732:
1726:
1716:
1714:
1713:
1710:
1707:
1696:
1690:
1684:
1675:
1671:
1665:
1663:
1659:Harmonices Mundi
1650:
1644:
1638:
1632:
1626:
1585:
1573:
1567:
1559:
1552:
1546:
1539:
1533:
1522:
1516:
1505:
1499:
1498:
1478:
1467:
1466:
1464:
1463:
1456:
1453:
1441:
1432:
1428:
1421:
1419:
1418:
1413:
1411:
1269:
1261:
1243:
1233:integer sequence
1224:
1193:
1181:
1180:
1178:
1177:
1174:
1171:
1150:
1142:
1141:
1139:
1138:
1135:
1132:
1114:
1110:
1097:
1096:
1094:
1093:
1090:
1087:
1065:
1064:
1062:
1061:
1058:
1055:
1032:
1019:
1007:
991:
980:
970:
964:
952:
950:
949:
944:
942:
940:
933:
932:
917:
916:
897:
892:
891:
861:
859:
858:
855:
852:
845:
843:
842:
839:
836:
829:
827:
826:
823:
820:
813:
811:
810:
807:
804:
797:
795:
794:
791:
788:
781:
779:
778:
775:
772:
765:
763:
762:
759:
756:
749:
747:
746:
743:
740:
733:
731:
730:
727:
724:
717:
715:
714:
711:
708:
701:
699:
698:
695:
692:
685:
683:
682:
679:
676:
669:
667:
666:
663:
660:
653:
651:
650:
647:
644:
599:
598:
596:
595:
590:
587:
573:
572:
570:
569:
564:
561:
551:
537:
536:
534:
533:
524:
521:
511:
510:
508:
507:
502:
499:
486:
485:
483:
482:
477:
474:
460:
459:
457:
456:
451:
448:
439:, the parent of
438:
436:
434:
433:
428:
425:
415:
414:
412:
411:
402:
399:
389:
388:
386:
385:
380:
377:
368:, the parent of
367:
365:
363:
362:
357:
354:
344:
338:
328:
327:
325:
324:
319:
316:
303:
302:
300:
299:
294:
291:
255:Harmonices Mundi
218:Harmonices Mundi
178:Harmonices Mundi
163:
162:
160:
159:
154:
151:
137:
136:
134:
133:
124:
121:
111:
110:
108:
107:
102:
99:
85:rational numbers
66:Calkin–Wilf tree
50:
38:
21:
2265:
2264:
2260:
2259:
2258:
2256:
2255:
2254:
2235:
2234:
2217:
2199:
2198:
2180:
2179:
2176:
2149:
2109:
2071:Gibbons, Jeremy
2069:
2056:10.2307/3647762
2037:
2023:
2013:Springer-Verlag
1999:
1959:
1945:10.2307/2589182
1922:
1913:
1889:
1855:
1850:
1828:
1825:
1820:
1819:
1811:
1807:
1796:
1788:
1782:
1778:
1770:
1766:
1758:
1754:
1745:
1741:
1733:
1729:
1711:
1708:
1705:
1704:
1702:
1697:
1693:
1685:
1678:
1672:
1668:
1652:
1651:
1647:
1639:
1635:
1627:
1623:
1618:
1592:
1584:fusc(6 + 1) = 3
1583:
1569:
1561:
1554:
1548:
1541:
1535:
1528:
1518:
1507:
1504:
1503:
1497:
1492:
1491:
1490:
1489:
1488:
1484:
1472:
1457:
1454:
1447:
1446:
1444:
1443:
1437:
1430:
1426:
1409:
1408:
1359:
1332:
1331:
1309:
1282:
1281:
1263:
1257:
1239:
1222:
1212:
1197:
1191:
1175:
1172:
1169:
1168:
1166:
1163:
1158:
1154:
1148:
1136:
1133:
1130:
1129:
1127:
1124:
1119:
1112:
1108:
1103:
1091:
1088:
1085:
1084:
1082:
1080:
1074:
1072:
1059:
1056:
1053:
1052:
1050:
1048:
1042:
1040:
1030:
1025:
1017:
1005:
1000:
992:represents the
988:
982:
978:
972:
966:
962:
957:
924:
908:
901:
877:
872:
871:
856:
853:
850:
849:
847:
840:
837:
834:
833:
831:
824:
821:
818:
817:
815:
808:
805:
802:
801:
799:
792:
789:
786:
785:
783:
776:
773:
770:
769:
767:
760:
757:
754:
753:
751:
744:
741:
738:
737:
735:
728:
725:
722:
721:
719:
712:
709:
706:
705:
703:
696:
693:
690:
689:
687:
680:
677:
674:
673:
671:
664:
661:
658:
657:
655:
648:
645:
642:
641:
639:
621:
591:
588:
579:
578:
576:
575:
565:
562:
557:
556:
554:
553:
539:
525:
522:
517:
516:
514:
513:
503:
500:
495:
494:
492:
491:
478:
475:
466:
465:
463:
462:
452:
449:
444:
443:
441:
440:
429:
426:
421:
420:
418:
417:
403:
400:
395:
394:
392:
391:
381:
378:
373:
372:
370:
369:
358:
355:
350:
349:
347:
346:
340:
334:
320:
317:
312:
311:
309:
308:
295:
292:
287:
286:
284:
283:
264:
206:
155:
152:
143:
142:
140:
139:
125:
122:
117:
116:
114:
113:
103:
100:
95:
94:
92:
91:
58:
57:
56:
55:
54:
51:
43:
42:
39:
28:
23:
22:
15:
12:
11:
5:
2263:
2261:
2253:
2252:
2247:
2237:
2236:
2233:
2232:
2215:
2196:
2175:
2174:External links
2172:
2171:
2170:
2147:
2120:(4): 265–283,
2107:
2087:(3): 281–291,
2067:
2050:(7): 642–643,
2035:
2021:
1997:
1977:(2): 275–278,
1957:
1914:Calkin, Neil;
1911:
1887:
1853:
1848:
1830:Aigner, Martin
1824:
1821:
1818:
1817:
1805:
1795:instead of by
1785:Carlitz (1964)
1776:
1772:Carlitz (1964)
1764:
1752:
1739:
1727:
1691:
1676:
1666:
1645:
1633:
1620:
1619:
1617:
1614:
1591:
1588:
1580:
1579:
1501:
1500:
1493:
1486:
1485:
1423:
1422:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1360:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1310:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1289:
1250:
1249:
1223:fusc(0...4096)
1211:
1208:
1200:
1199:
1195:
1161:
1156:
1152:
1122:
1106:
1100:
1099:
1078:
1070:
1067:
1046:
1038:
1028:
1003:
986:
976:
960:
954:
953:
939:
936:
931:
927:
923:
920:
915:
911:
907:
904:
900:
895:
890:
887:
884:
880:
864:
863:
620:
617:
280:directed graph
263:
260:
212:The tree from
205:
202:
52:
45:
44:
40:
33:
32:
31:
30:
29:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2262:
2251:
2248:
2246:
2243:
2242:
2240:
2230:
2226:
2225:
2220:
2216:
2212:
2211:
2206:
2202:
2197:
2193:
2192:
2187:
2183:
2178:
2177:
2173:
2166:
2162:
2161:
2156:
2152:
2148:
2145:
2141:
2136:
2131:
2127:
2123:
2119:
2115:
2114:
2108:
2104:
2100:
2095:
2090:
2086:
2082:
2081:
2076:
2075:Bird, Richard
2072:
2068:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2036:
2033:
2029:
2024:
2022:0-387-90652-5
2018:
2014:
2009:
2008:
2002:
1998:
1994:
1990:
1985:
1980:
1976:
1972:
1971:
1966:
1962:
1958:
1954:
1950:
1946:
1942:
1938:
1934:
1930:
1929:
1921:
1917:
1916:Wilf, Herbert
1912:
1908:
1903:
1899:
1895:
1894:
1888:
1885:
1881:
1876:
1871:
1867:
1863:
1859:
1854:
1851:
1845:
1841:
1840:
1835:
1831:
1827:
1826:
1822:
1814:
1809:
1806:
1800:
1792:
1786:
1780:
1777:
1773:
1768:
1765:
1761:
1756:
1753:
1749:
1743:
1740:
1736:
1731:
1728:
1724:
1720:
1700:
1695:
1692:
1688:
1683:
1681:
1677:
1670:
1667:
1661:
1660:
1655:
1649:
1646:
1642:
1637:
1634:
1630:
1625:
1622:
1615:
1613:
1611:
1607:
1602:
1597:
1589:
1587:
1577:
1576:
1575:
1572:
1565:
1557:
1551:
1544:
1538:
1532:
1526:
1525:powers of two
1521:
1515:
1511:
1496:
1482:
1476:
1471:The function
1469:
1461:
1451:
1440:
1434:
1405:
1399:
1396:
1393:
1387:
1384:
1381:
1375:
1369:
1366:
1363:
1361:
1353:
1350:
1347:
1344:
1338:
1335:
1325:
1319:
1316:
1313:
1311:
1303:
1300:
1294:
1291:
1280:
1279:
1278:
1277:
1273:
1272:fusc function
1267:
1260:
1255:
1247:
1242:
1237:
1236:
1235:
1234:
1230:
1220:
1216:
1207:
1205:
1189:
1185:
1164:
1157:
1146:
1125:
1118:
1117:
1116:
1109:
1077:
1068:
1045:
1036:
1035:
1034:
1031:
1023:
1015:
1011:
1006:
997:
995:
994:integral part
989:
975:
969:
963:
937:
934:
929:
925:
921:
913:
909:
902:
898:
893:
888:
885:
882:
878:
870:
869:
868:
638:
637:
636:
634:
625:
618:
616:
614:
610:
606:
601:
594:
586:
582:
568:
560:
550:
546:
542:
532:
528:
520:
506:
498:
488:
481:
473:
469:
455:
447:
432:
424:
410:
406:
398:
384:
376:
361:
353:
343:
337:
332:
323:
315:
305:
298:
290:
281:
273:
268:
261:
259:
257:
256:
252:
248:
244:
240:
236:
232:
228:
220:
219:
215:
210:
203:
201:
199:
198:fusc function
195:
191:
187:
182:
180:
179:
175:
171:
167:
158:
150:
146:
132:
128:
120:
106:
98:
90:
86:
83:
79:
75:
72:in which the
71:
67:
63:
62:number theory
49:
37:
19:
2229:Cut-the-knot
2223:
2208:
2189:
2164:
2158:
2117:
2111:
2084:
2078:
2047:
2043:
2006:
1974:
1968:
1932:
1926:
1897:
1891:
1865:
1861:
1837:
1808:
1798:
1790:
1779:
1767:
1762:, Theorem 1.
1755:
1742:
1730:
1694:
1669:
1658:
1648:
1641:Raney (1973)
1636:
1631:, Section 6.
1624:
1593:
1581:
1570:
1563:
1555:
1549:
1542:
1536:
1530:
1523:as a sum of
1519:
1513:
1509:
1494:
1483:of the form
1474:
1470:
1459:
1449:
1438:
1435:
1424:
1271:
1265:
1258:
1251:
1228:
1227:
1201:
1159:
1120:
1104:
1101:
1075:
1043:
1026:
1001:
998:
984:
973:
967:
965:denotes the
958:
955:
865:
632:
630:
602:
592:
584:
580:
566:
558:
548:
544:
540:
530:
526:
518:
504:
496:
489:
479:
471:
467:
453:
445:
430:
422:
408:
404:
396:
382:
374:
359:
351:
341:
335:
321:
313:
306:
296:
288:
277:
253:
238:
235:Jean Berstel
231:Herbert Wilf
224:
216:
197:
193:
189:
183:
176:
170:Herbert Wilf
156:
148:
144:
130:
126:
118:
104:
96:
65:
59:
1961:Carlitz, L.
1939:: 360–363,
1431:fusc(1) = 1
1427:fusc(0) = 0
1219:Scatterplot
227:Neil Calkin
166:Neil Calkin
76:correspond
2239:Categories
1823:References
1654:Kepler, J.
1188:odd number
1147:is hence
1033:.Example:
329:for which
239:Raney tree
78:one-to-one
2210:MathWorld
2191:MathWorld
2167:: 193–220
2144:120933574
1388:
1370:
1339:
1320:
1295:
922:−
919:⌋
906:⌊
345:is 1. If
2153:(1858),
2103:14237968
2003:(1982),
1918:(2000),
1836:(2004),
1656:(1619),
1553:and for
258:(1619).
251:Kepler's
214:Kepler's
174:Kepler's
89:fraction
82:positive
74:vertices
2064:3647762
1993:0157907
1953:2589182
1884:2673006
1715:
1703:
1465:
1445:
1270:of the
1244:in the
1241:A002487
1231:is the
1179:
1167:
1140:
1128:
1095:
1083:
1063:
1051:
1012:of the
860:
848:
844:
832:
828:
816:
812:
800:
796:
784:
780:
768:
764:
752:
748:
736:
732:
720:
716:
704:
700:
688:
684:
672:
668:
656:
652:
640:
597:
577:
571:
555:
535:
515:
509:
493:
484:
464:
458:
442:
435:
419:
413:
393:
387:
371:
364:
348:
326:
310:
301:
285:
274:layout.
204:History
161:
141:
135:
115:
109:
93:
80:to the
2142:
2101:
2062:
2019:
1991:
1951:
1882:
1846:
1256:, the
1252:Using
1194:= 1001
1182:: The
1151:= 1110
1143:: The
981:, and
956:where
437:> 1
366:< 1
272:H tree
221:(1619)
64:, the
2140:S2CID
2099:S2CID
2060:JSTOR
1949:JSTOR
1935:(4),
1923:(PDF)
1797:fusc(
1789:fusc(
1616:Notes
1512:<
1508:0 ≤ 2
1473:fusc(
1458:fusc(
1448:fusc(
1264:fusc(
1155:= 14.
547:>
416:; if
68:is a
2017:ISBN
1844:ISBN
1801:+ 1)
1477:+ 1)
1462:+ 1)
1436:The
1429:and
1385:fusc
1367:fusc
1336:fusc
1317:fusc
1292:fusc
1246:OEIS
1198:= 9.
1079:1990
1047:1081
862:, ….
631:The
339:and
229:and
168:and
138:and
70:tree
2130:hdl
2122:doi
2118:206
2089:doi
2052:doi
2048:110
1979:doi
1967:",
1941:doi
1933:107
1902:doi
1898:178
1870:doi
1566:+ 1
1558:− 1
1545:− 1
1221:of
1016:of
979:= 1
461:is
390:is
333:of
60:In
2241::
2227:,
2221:,
2207:,
2203:,
2188:,
2184:,
2165:55
2163:,
2157:,
2138:,
2128:,
2116:,
2097:,
2085:16
2083:,
2058:,
2046:,
2042:,
2026:.
2015:,
2011:,
1989:MR
1987:,
1975:70
1973:,
1947:,
1931:,
1925:,
1896:,
1880:MR
1878:,
1866:31
1864:,
1860:,
1832:;
1679:^
1586:.
1506:,
1433:.
1248:).
1165:=
1126:=
1092:53
1086:37
1081:=
1060:37
1054:53
1049:=
996:.
983:⌊
846:,
830:,
814:,
798:,
782:,
766:,
750:,
734:,
718:,
702:,
686:,
670:,
654:,
615:.
600:.
583:+
543:+
529:+
470:−
407:−
200:.
181:.
147:+
129:+
2169:.
2132::
2124::
2106:.
2091::
2054::
1996:.
1981::
1956:.
1943::
1904::
1872::
1815:.
1803:.
1799:n
1793:)
1791:n
1774:.
1712:1
1709:/
1706:1
1689:.
1664:.
1643:.
1571:n
1564:n
1562:2
1556:n
1550:n
1543:n
1537:n
1531:n
1529:2
1520:n
1514:n
1510:r
1502:)
1495:r
1487:(
1475:n
1460:n
1455:/
1452:)
1450:n
1439:n
1406:,
1403:)
1400:1
1397:+
1394:n
1391:(
1382:+
1379:)
1376:n
1373:(
1364:=
1357:)
1354:1
1351:+
1348:n
1345:2
1342:(
1329:)
1326:n
1323:(
1314:=
1307:)
1304:n
1301:2
1298:(
1268:)
1266:n
1259:n
1196:2
1192:i
1176:3
1173:/
1170:4
1162:i
1160:q
1153:2
1149:i
1137:4
1134:/
1131:3
1123:i
1121:q
1113:i
1107:i
1105:q
1098:.
1089:/
1076:q
1071:2
1066:.
1057:/
1044:q
1039:2
1029:i
1027:q
1018:i
1004:i
1002:q
990:⌋
987:i
985:q
977:1
974:q
968:i
961:i
959:q
938:1
935:+
930:i
926:q
914:i
910:q
903:2
899:1
894:=
889:1
886:+
883:i
879:q
857:4
854:/
851:3
841:3
838:/
835:5
825:5
822:/
819:2
809:2
806:/
803:5
793:5
790:/
787:3
777:3
774:/
771:4
761:4
758:/
755:1
745:1
742:/
739:3
729:3
726:/
723:2
713:2
710:/
707:3
697:3
694:/
691:1
681:1
678:/
675:2
665:2
662:/
659:1
649:1
646:/
643:1
593:b
589:/
585:b
581:a
567:b
563:/
559:a
549:a
545:b
541:a
531:b
527:a
523:/
519:a
505:b
501:/
497:a
480:b
476:/
472:b
468:a
454:b
450:/
446:a
431:b
427:/
423:a
409:a
405:b
401:/
397:a
383:b
379:/
375:a
360:b
356:/
352:a
342:b
336:a
322:b
318:/
314:a
297:b
293:/
289:a
157:b
153:/
149:b
145:a
131:b
127:a
123:/
119:a
105:b
101:/
97:a
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
