537:
Applications of
Fibonacci Numbers (Proceedings of the Fourth International Conference on Fibonacci Numbers and Their Applications, Wake Forest University, N.C., U.S.A., July 30–August 3, 1990)
534:
Fielder, Daniel C.; Alford, Cecil O. (1991), "Pascal's triangle: Top gun or just one of the gang?", in Bergum, Gerald E.; Philippou, Andreas N.; Horadam, A. F. (eds.),
497:
Rota Bulò, Samuel; Hancock, Edwin R.; Aziz, Furqan; Pelillo, Marcello (2012), "Efficient computation of Ihara coefficients using the Bell polynomial recursion",
143:
Arrays in which the length of each row grows as a linear function of the row number (rather than being equal to the row number) have also been considered.
27:
41:
of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index. That is, the
545:
447:
339:
234:
674:
598:
Millar, Jessica; Sloane, N. J. A.; Young, Neal E. (1996), "A new operation on sequences: the
Boustrouphedon transform",
102:
124:
Triangular arrays of integers in which each row is symmetric and begins and ends with 1 are sometimes called
360:
162:
70:
118:
75:
479:
114:
93:
151:
87:
81:
128:; examples include Pascal's triangle, the Narayana numbers, and the triangle of Eulerian numbers.
625:
607:
580:
304:
269:
243:
66:
647:
541:
443:
335:
178:
155:
108:
20:
535:
617:
570:
506:
393:
296:
253:
166:
137:
97:
520:
424:
316:
265:
215:
516:
420:
312:
261:
211:
198:
464:
483:
668:
229:
202:
62:
651:
629:
584:
287:
Velleman, Daniel J.; Call, Gregory S. (1995), "Permutations and combination locks",
273:
136:
Triangular arrays may list mathematical values other than numbers; for instance the
355:
232:; Liese, Jeffrey (2013), "Harmonic numbers, Catalan's triangle and mesh patterns",
111:, counting strings of balanced parentheses with a given number of distinct nestings
31:
561:
Thacher Jr., Henry C. (July 1964), "Remark on
Algorithm 60: Romberg integration",
26:
78:, which counts strings of parentheses in which no close parenthesis is unmatched
257:
511:
397:
656:
621:
575:
465:"On integer-sequence-based constructions of generalized Pascal triangles"
612:
308:
411:
Barry, Paul (2011), "On a generalization of the
Narayana triangle",
381:
300:
30:
The triangular array whose right-hand diagonal sequence consists of
140:
form a triangular array in which each array entry is a polynomial.
248:
181:, the number of entries in such an array up to some particular row
25:
440:
Pascal's
Arithmetical Triangle: The Story of a Mathematical Idea
332:
Programming by design: a first course in structured programming
330:
Miller, Philip L.; Miller, Lee W.; Jackson, Purvis M. (1987),
210:, Santa Clara, Calif.: Fibonacci Association, pp. 69–71,
204:
A collection of manuscripts related to the
Fibonacci sequence
84:, which counts permutations with a given number of ascents
382:"Die Isomerie-Arten bei den Homologen der Paraffin-Reihe"
158:by completing the values in a triangle of numbers.
90:, whose entries are all of the integers in order
105:, used in the mathematics of chemical compounds
8:
201:(1980), "A triangle for the Bell numbers",
57:Notable particular examples include these:
611:
574:
510:
247:
165:uses a triangular array to transform one
334:, Wadsworth Pub. Co., pp. 211–212,
69:in which a given element is the largest
190:
154:can be used to estimate the value of a
7:
16:Concept in mathematics and computing
499:Linear Algebra and Its Applications
14:
37:In mathematics and computing, a
600:Journal of Combinatorial Theory
358:(1976), "Fibonacci triangle",
1:
540:, Springer, pp. 77–90,
472:Journal of Integer Sequences
413:Journal of Integer Sequences
126:generalized Pascal triangles
691:
438:Edwards, A. W. F. (2002),
380:Losanitsch, S. M. (1897),
258:10.1016/j.disc.2013.03.017
65:, whose numbers count the
18:
563:Communications of the ACM
512:10.1016/j.laa.2011.08.017
419:(4): Article 11.4.5, 22,
398:10.1002/cber.189703002144
117:, whose entries are the
19:Not to be confused with
361:The Fibonacci Quarterly
163:Boustrophedon transform
622:10.1006/jcta.1996.0087
34:
576:10.1145/364520.364542
119:binomial coefficients
45:th row contains only
29:
675:Triangles of numbers
289:Mathematics Magazine
235:Discrete Mathematics
484:2006JIntS...9...24B
67:partitions of a set
648:Weisstein, Eric W.
463:Barry, P. (2006),
103:Lozanić's triangle
76:Catalan's triangle
35:
652:"Number Triangle"
242:(14): 1515–1531,
179:Triangular number
156:definite integral
115:Pascal's triangle
109:Narayana triangle
98:Fibonacci numbers
94:Hosoya's triangle
21:Triangular matrix
682:
661:
660:
634:
632:
615:
595:
589:
587:
578:
558:
552:
550:
531:
525:
523:
514:
505:(5): 1436–1441,
494:
488:
486:
469:
460:
454:
452:
435:
429:
427:
408:
402:
400:
392:(2): 1917–1926,
377:
371:
369:
352:
346:
344:
327:
321:
319:
284:
278:
276:
251:
226:
220:
218:
209:
199:Shallit, Jeffrey
195:
167:integer sequence
152:Romberg's method
138:Bell polynomials
88:Floyd's triangle
82:Euler's triangle
39:triangular array
690:
689:
685:
684:
683:
681:
680:
679:
665:
664:
646:
645:
642:
637:
613:math.CO/0205218
597:
596:
592:
560:
559:
555:
548:
533:
532:
528:
496:
495:
491:
478:(6.2.4): 1–34,
467:
462:
461:
457:
450:
437:
436:
432:
410:
409:
405:
379:
378:
374:
354:
353:
349:
342:
329:
328:
324:
301:10.2307/2690567
286:
285:
281:
228:
227:
223:
207:
197:
196:
192:
188:
175:
149:
134:
132:Generalizations
96:, based on the
55:
24:
17:
12:
11:
5:
688:
686:
678:
677:
667:
666:
663:
662:
641:
640:External links
638:
636:
635:
590:
569:(7): 420–421,
553:
546:
526:
489:
455:
448:
430:
403:
372:
347:
340:
322:
295:(4): 243–253,
279:
230:Kitaev, Sergey
221:
189:
187:
184:
183:
182:
174:
171:
169:into another.
148:
145:
133:
130:
122:
121:
112:
106:
100:
91:
85:
79:
73:
54:
51:
15:
13:
10:
9:
6:
4:
3:
2:
687:
676:
673:
672:
670:
659:
658:
653:
649:
644:
643:
639:
631:
627:
623:
619:
614:
609:
605:
601:
594:
591:
586:
582:
577:
572:
568:
564:
557:
554:
549:
547:9780792313090
543:
539:
538:
530:
527:
522:
518:
513:
508:
504:
500:
493:
490:
485:
481:
477:
473:
466:
459:
456:
451:
449:9780801869464
445:
442:, JHU Press,
441:
434:
431:
426:
422:
418:
414:
407:
404:
399:
395:
391:
387:
383:
376:
373:
367:
363:
362:
357:
356:Hosoya, Haruo
351:
348:
343:
341:9780534082444
337:
333:
326:
323:
318:
314:
310:
306:
302:
298:
294:
290:
283:
280:
275:
271:
267:
263:
259:
255:
250:
245:
241:
237:
236:
231:
225:
222:
217:
213:
206:
205:
200:
194:
191:
185:
180:
177:
176:
172:
170:
168:
164:
159:
157:
153:
146:
144:
141:
139:
131:
129:
127:
120:
116:
113:
110:
107:
104:
101:
99:
95:
92:
89:
86:
83:
80:
77:
74:
72:
68:
64:
63:Bell triangle
60:
59:
58:
52:
50:
48:
44:
40:
33:
28:
22:
655:
606:(1): 44–54,
603:
602:, Series A,
599:
593:
566:
562:
556:
536:
529:
502:
498:
492:
475:
471:
458:
439:
433:
416:
412:
406:
389:
385:
375:
368:(2): 173–178
365:
359:
350:
331:
325:
292:
288:
282:
239:
233:
224:
203:
193:
160:
150:
147:Applications
142:
135:
125:
123:
56:
46:
42:
38:
36:
32:Bell numbers
386:Chem. Ber.
186:References
49:elements.
657:MathWorld
249:1209.6423
71:singleton
669:Category
630:15637402
585:29898282
274:18248485
173:See also
53:Examples
521:2890929
480:Bibcode
425:2792161
317:1363707
309:2690567
266:3047390
216:0624091
628:
583:
544:
519:
446:
423:
338:
315:
307:
272:
264:
214:
626:S2CID
608:arXiv
581:S2CID
468:(PDF)
305:JSTOR
270:S2CID
244:arXiv
208:(PDF)
542:ISBN
444:ISBN
336:ISBN
161:The
61:The
618:doi
571:doi
507:doi
503:436
394:doi
297:doi
254:doi
240:313
671::
654:,
650:,
624:,
616:,
604:76
579:,
565:,
517:MR
515:,
501:,
474:,
470:,
421:MR
417:14
415:,
390:30
388:,
384:,
366:14
364:,
313:MR
311:,
303:,
293:68
291:,
268:,
262:MR
260:,
252:,
238:,
212:MR
633:.
620::
610::
588:.
573::
567:7
551:.
524:.
509::
487:.
482::
476:9
453:.
428:.
401:.
396::
370:.
345:.
320:.
299::
277:.
256::
246::
219:.
47:i
43:i
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
