27:
152:
146:
988:
140:
The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue).
405:
534:
693:, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971, … (sequence
266:
614:
791:
56:
783:
135:
1057:
737:
700:
117:
This is also the number of points of a hexagonal lattice with nearest-neighbor coupling whose distance from a given point is less than or equal to
413:
Each centered triangular number has a remainder of 1 when divided by 3, and the quotient (if positive) is the previous regular triangular number.
295:
1011:
749:
If the centered triangular numbers are treated as the coefficients of the McLaurin series of a function, that function converges for all
78:
1050:
1235:
39:
1240:
1220:
466:
49:
43:
35:
114:
with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers.
1230:
1212:
1111:
1101:
145:
1410:
1335:
1126:
1121:
1116:
1106:
1083:
1043:
60:
983:{\displaystyle 1+4x+10x^{2}+19x^{3}+31x^{4}+~...={\frac {1-x^{3}}{(1-x)^{4}}}={\frac {x^{2}+x+1}{(1-x)^{3}}}~.}
189:
545:
1301:
1278:
1203:
1315:
1096:
151:
1370:
1225:
111:
1268:
1074:
1296:
1273:
1189:
1263:
1250:
1169:
1159:
1149:
1021:
1018:
1007:
752:
419:
1330:
1288:
1184:
1179:
1174:
1164:
1141:
1066:
999:
460:
The centered triangular numbers can be expressed in terms of the centered square numbers:
169:
107:
104:
418:
Each centered triangular number from 10 onwards is the sum of three consecutive regular
1258:
436:
120:
1404:
1380:
1154:
730:
1, 10, 136, 1 891, 26 335, 366 796, 5 108 806, 71 156 485, 991 081 981, … (sequence
1385:
1340:
690:
686:
682:
678:
674:
670:
448:
1375:
1131:
666:
662:
658:
654:
650:
646:
642:
638:
1355:
1026:
16:
Centered figurate number that represents a triangle with a dot in the center
785:, in which case it can be expressed as the meromorphic generating function
156:
400:{\displaystyle C_{3,n}=1+3{\frac {n(n+1)}{2}}={\frac {3n^{2}+3n+2}{2}}.}
707:
The first simultaneously triangular and centered triangular numbers (
1006:(1936), republished by W. W. Norton & Company (September 1993),
1035:
150:
1039:
20:
732:
695:
176:-th centered triangular number, corresponding to the (
794:
755:
548:
469:
298:
192:
123:
1363:
1353:
1323:
1314:
1287:
1249:
1211:
1202:
1140:
1082:
1073:
982:
777:
608:
528:
399:
260:
129:
278:-th centered triangular number, corresponding to
155:The first eight centered triangular numbers on a
529:{\displaystyle C_{3,n}={\frac {3C_{4,n}+1}{4}},}
48:but its sources remain unclear because it lacks
1051:
8:
1360:
1320:
1208:
1079:
1058:
1044:
1036:
965:
929:
922:
910:
886:
873:
849:
833:
817:
793:
764:
756:
754:
597:
572:
553:
547:
499:
489:
474:
468:
456:Relationship with centered square numbers
367:
357:
327:
303:
297:
261:{\displaystyle C_{3,n+1}-C_{3,n}=3(n+1).}
222:
197:
191:
122:
79:Learn how and when to remove this message
609:{\displaystyle C_{4,n}=n^{2}+(n+1)^{2}.}
624:The first centered triangular numbers (
7:
620:Lists of centered triangular numbers
286:the center, is given by the formula:
435:centered triangular numbers is the
14:
144:
25:
962:
949:
907:
894:
765:
757:
594:
581:
345:
333:
252:
240:
1:
1236:Centered dodecahedral numbers
431:> 2, the sum of the first
180:+ 1)-th triangular layer, is:
1241:Centered icosahedral numbers
1221:Centered tetrahedral numbers
1022:"Centered Triangular Number"
1231:Centered octahedral numbers
1112:Centered heptagonal numbers
1102:Centered pentagonal numbers
1092:Centered triangular numbers
1004:Mathematics for the Million
1427:
1336:Squared triangular numbers
1127:Centered decagonal numbers
1122:Centered nonagonal numbers
1117:Centered octagonal numbers
1107:Centered hexagonal numbers
1302:Square pyramidal numbers
1279:Stella octangula numbers
778:{\displaystyle |x|<1}
34:This article includes a
1097:Centered square numbers
745:The generating function
63:more precise citations.
984:
779:
610:
530:
401:
262:
159:
131:
1226:Centered cube numbers
985:
780:
611:
531:
402:
263:
154:
132:
1269:Dodecahedral numbers
792:
753:
546:
467:
296:
190:
121:
112:equilateral triangle
1386:8-hypercube numbers
1381:7-hypercube numbers
1376:6-hypercube numbers
1371:5-hypercube numbers
1341:Tesseractic numbers
1297:Tetrahedral numbers
1274:Icosahedral numbers
1190:Dodecagonal numbers
110:that represents an
1264:Octahedral numbers
1170:Heptagonal numbers
1160:Pentagonal numbers
1150:Triangular numbers
1019:Weisstein, Eric W.
980:
775:
606:
526:
420:triangular numbers
397:
258:
160:
127:
36:list of references
1398:
1397:
1394:
1393:
1349:
1348:
1331:Pentatope numbers
1310:
1309:
1198:
1197:
1185:Decagonal numbers
1180:Nonagonal numbers
1175:Octagonal numbers
1165:Hexagonal numbers
1012:978-0-393-31071-9
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521:
392:
352:
130:{\displaystyle n}
101:triangular number
89:
88:
81:
1418:
1411:Figurate numbers
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1321:
1209:
1080:
1067:Figurate numbers
1060:
1053:
1046:
1037:
1032:
1031:
989:
987:
986:
981:
974:
973:
971:
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969:
947:
934:
933:
923:
918:
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915:
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892:
891:
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853:
838:
837:
822:
821:
784:
782:
781:
776:
768:
760:
735:
698:
634:< 3000) are:
615:
613:
612:
607:
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601:
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576:
564:
563:
535:
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532:
527:
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406:
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403:
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372:
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328:
314:
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267:
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259:
233:
232:
214:
213:
148:
136:
134:
133:
128:
84:
77:
73:
70:
64:
59:this article by
50:inline citations
29:
28:
21:
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1000:Lancelot Hogben
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593:
568:
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363:
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329:
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218:
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165:
119:
118:
108:figurate number
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68:
65:
54:
40:related reading
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26:
17:
12:
11:
5:
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1196:
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1192:
1187:
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1162:
1157:
1155:Square numbers
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1138:
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806:
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800:
797:
774:
771:
767:
763:
759:
746:
743:
742:
741:
726:< 10) are:
721:
711:
705:
704:
628:
621:
618:
617:
616:
605:
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586:
583:
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513:
508:
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502:
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494:
488:
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457:
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453:
452:
437:magic constant
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423:
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288:
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126:
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44:external links
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1317:
1316:4-dimensional
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1218:
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1207:
1205:
1204:3-dimensional
1201:
1191:
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1103:
1100:
1098:
1095:
1093:
1090:
1089:
1087:
1085:
1081:
1078:
1076:
1075:2-dimensional
1072:
1068:
1061:
1056:
1054:
1049:
1047:
1042:
1041:
1038:
1029:
1028:
1023:
1020:
1015:
1013:
1009:
1005:
1001:
998:
997:
993:
977:
966:
958:
955:
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944:
941:
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935:
930:
926:
919:
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903:
900:
897:
887:
883:
879:
876:
870:
867:
864:
861:
855:
850:
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842:
839:
834:
830:
826:
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814:
810:
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804:
801:
798:
795:
788:
787:
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772:
769:
761:
744:
739:
734:
729:
728:
727:
724:
720:
715:
710:
702:
697:
692:
688:
684:
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676:
672:
668:
664:
660:
656:
652:
648:
644:
640:
637:
636:
635:
632:
627:
619:
603:
598:
590:
587:
584:
578:
573:
569:
565:
560:
557:
554:
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542:
541:
540:
523:
518:
514:
511:
506:
503:
500:
496:
492:
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481:
478:
475:
471:
463:
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461:
455:
450:
446:
442:
438:
434:
430:
426:
425:
421:
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416:
412:
411:
394:
389:
385:
382:
379:
376:
373:
368:
364:
360:
354:
349:
342:
339:
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321:
318:
315:
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304:
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292:
291:
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281:
277:
273:
272:
255:
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246:
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229:
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215:
210:
207:
204:
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198:
194:
186:
185:
184:
183:
179:
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171:
167:
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162:
158:
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149:
147:
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124:
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98:
94:
83:
80:
72:
62:
58:
52:
51:
45:
41:
37:
32:
23:
22:
19:
1364:non-centered
1324:non-centered
1259:Cube numbers
1251:non-centered
1142:non-centered
1132:Star numbers
1091:
1025:
1003:
748:
722:
718:
713:
708:
706:
630:
625:
623:
538:
459:
449:magic square
444:
440:
432:
428:
283:
279:
275:
177:
173:
143:
139:
116:
100:
96:
92:
90:
75:
66:
55:Please help
47:
18:
1356:dimensional
61:introducing
994:References
163:Properties
1289:pyramidal
1027:MathWorld
956:−
901:−
880:−
216:−
69:June 2014
1405:Category
1213:centered
1084:centered
157:hex grid
105:centered
93:centered
1354:Higher
736:in the
733:A128862
699:in the
696:A005448
447:normal
439:for an
282:layers
172:of the
97:centred
57:improve
1010:
975:
859:
539:where
170:gnomon
103:is a
42:, or
1008:ISBN
770:<
738:OEIS
701:OEIS
427:For
284:plus
274:The
168:The
95:(or
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443:by
1407::
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
