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Repeating the aforementioned procedure on this modified Pascal system gives the Lucas numbers in one direction, and the
Fibonacci in the other. In fact any Fibonacci-like sequence, leading to the Golden Ratio, can be created by adjusting the sides of the triangle, and the 'seeds' of these sequences are in fact the numbers on the sides. So for the classical Pascal Triangle, with 1,1 on the sides, gives (1,1),2,3,5,8... and you can see that ..(1,2),3,5,8.. leads to a similar, but offset Fib. In the other direction (2,1),3,4,7,11... you get Lucas. This may seem like original research (it was for me) but I have to believe that mathematicians have known about this for ages, and there must be published sources one could hang the donkey tail onto so as to be able to include these facts into the main articles. Anyone know of any? Thanks.
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194:. Lucas sequences encompass not just Lucas numbers but also Fibonacci numbers, Pell numbers, and in fact any sequence defined by a linear recurrence relation with a quadratic characteristic equation. Making Lucas numbers a special case by merging them into the Lucas sequence article would be anomalous and misleading.
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I have edited your comment for readability. Three of these identities are in the article already (the second is the same as the first). The other one does not involve
Fibonacci numbers, but you placed it in the section titled "Relationship to Fibonacci numbers". It can also easily be derived from
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This article is about a specific important sequence. It is simple and easy to understand. The other article is very abstract and difficult to understand and is about many sequences, some of which may not be so important as this one. However, we should have a link to it as a generalization. In other
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There should not be credit given to lucas for realizing that just because you start somewhere else the aplied formula is still the applied formula and does not change or add to the integral knowledge that all here inlies One sequence of of liner occurance in relation to a quadratic equation, im
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In the Wiki article on the Pascal
Triangle there is a subsection showing how the Fibonacci numbers can be generated by summing samplings taken across every other diagonal. I've seen, online (but can't remember URL), an analogue to the Pascal triangle with one of the 1's sides replaced by 2's.
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not only encompass a broad range of famous integer sequences; they're also useful in applied mathematics (pseudo-primality testing, etc).
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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disagreeing with
Gandalf61, lucas is nothing special out of fibbonacci numbers all lucas preposals are credited out of context
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what this article also needs is a formula that can be used to calculate the nth term in the lucas numbers sequence!!!!!!
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1591:{\displaystyle F_{n}:={\begin{cases}{L_{n+m}+L_{n-m} \over {L_{m+1}+L_{m-1}}}&{\text{if m is uneven and }}m=: -->
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Next identities are very usefull if you need to simplify some complex fibs expressions. Besides, they are elegant.
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Hereby I want to send some generalisation formulas for two Items in the chapter
Relationship to Fibonacci numbers.
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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https://web.archive.org/web/20051126021243/http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucasNbs.html
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https://web.archive.org/web/20070216024906/http://www.plenilune.pwp.blueyonder.co.uk/fibonacci-calculator.asp
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1145:{\displaystyle F_{n}:={\begin{cases}{L_{n+m}+L_{n-m} \over {5F_{m}}}&{\text{if m is uneven and }}m=: -->
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1339:{\displaystyle F_{n}F_{m}:={\begin{cases}{L_{n+m}+L_{n-m} \over 5}&{\text{if m is uneven and }}m=: -->
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811:{\displaystyle L_{n}:={\begin{cases}{F_{n-m}+F_{n+m} \over F_{m}}&{\text{if m is uneven and }}m=: -->
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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The link "A Tutorial on
Generalized Lucas Numbers" is no longer valid. Anybody knows a new location?
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https://web.archive.org/web/20051030021553/http://milan.milanovic.org/math/english/lucas/lucas.html
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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before doing mass systematic removals. This message is updated dynamically through the template
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1982:{\displaystyle L_{n+k}+(-1)^{k}L_{n-k}=L_{k}L_{n},\quad L_{n+k}-(-1)^{k}L_{n-k}=5F_{n}F_{k}}
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1802:{\displaystyle F_{n+k}+(-1)^{k}F_{n-k}=L_{k}F_{n},\quad F_{n+k}-(-1)^{k}F_{n-k}=L_{n}F_{k}}
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https://web.archive.org/web/20130825030815/http://nakedprogrammer.com/LucasNumbers.aspx
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If you found an error with any archives or the URLs themselves, you can fix them with
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1;\\{L_{n+m}-L_{n-m} \over {L_{m+1}+L_{m-1}}}&{\text{if m is even and }}m=: -->
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1;\\{L_{n+m}-L_{n-m} \over {L_{m+1}+L_{m-1}}}&{\text{if m is even and }}m=: -->
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1;\\{L_{n+m}-L_{n-m} \over {L_{m+1}+L_{m-1}}}&{\text{if m is even and }}m=: -->
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944:{\displaystyle \,F_{n}={L_{n-1}+L_{n+1} \over 5}={L_{n-3}+L_{n+3} \over 10}}
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1;\\{L_{n+m}-L_{n-m} \over {5F_{m}}}&{\text{if m is even and }}m=: -->
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1;\\{L_{n+m}-L_{n-m} \over {5F_{m}}}&{\text{if m is even and }}m=: -->
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1;\\{L_{n+m}-L_{n-m} \over {5F_{m}}}&{\text{if m is even and }}m=: -->
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1;\\{F_{n-m}-F_{n+m} \over F_{m}}&{\text{if m is even and }}m=: -->
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1;\\{F_{n-m}-F_{n+m} \over F_{m}}&{\text{if m is even and }}m=: -->
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1;\\{F_{n-m}-F_{n+m} \over F_{m}}&{\text{if m is even and }}m=: -->
620:{\displaystyle \,L_{n}=F_{n-1}+F_{n+1}=F_{n}+2F_{n-1}=F_{n+2}-F_{n-2}}
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deserve their own article just as much as
Fibonacci numbers do. And
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1;\\{L_{n+m}-L_{n-m} \over 5}&{\text{if m is even and }}m=: -->
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1;\\{L_{n+m}-L_{n-m} \over 5}&{\text{if m is even and }}m=: -->
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http://www.plenilune.pwp.blueyonder.co.uk/fibonacci-calculator.asp
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where m is de position away from the Lucas number to find.
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for additional information. I made the following changes:
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Lucas numbers generated by Pascal
Triangle analogue
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354:Hello fellow Wikipedians,
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1600:2\\\end{cases}}}" /: -->
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1159:which we can rewrite as
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820:2\\\end{cases}}}" /: -->
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75:
67:
55:
54:
48:
37:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2050:
2039:
2036:
2034:
2031:
2030:
2028:
2019:
2015:
2011:
2006:
2005:
2004:
2003:
1999:
1995:
1974:
1970:
1964:
1960:
1956:
1953:
1948:
1945:
1942:
1938:
1932:
1924:
1921:
1915:
1910:
1907:
1904:
1900:
1894:
1889:
1885:
1879:
1875:
1871:
1866:
1863:
1860:
1856:
1850:
1842:
1839:
1833:
1828:
1825:
1822:
1818:
1810:
1794:
1790:
1784:
1780:
1776:
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1768:
1765:
1761:
1755:
1747:
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1738:
1733:
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1641:
1633:
1632:
1631:
1625:
1623:
1621:
1617:
1613:
1612:86.84.102.126
1609:
1578:
1575:
1572:
1557:
1554:
1551:
1547:
1543:
1538:
1535:
1532:
1528:
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1514:
1510:
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1501:
1498:
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1480:
1477:
1474:
1471:
1456:
1453:
1450:
1446:
1442:
1437:
1434:
1431:
1427:
1419:
1416:
1413:
1409:
1405:
1400:
1397:
1394:
1390:
1380:
1375:
1370:
1366:
1356:
1355:
1354:
1326:
1323:
1320:
1308:
1302:
1299:
1296:
1292:
1288:
1283:
1280:
1277:
1273:
1262:
1259:
1256:
1253:
1241:
1235:
1232:
1229:
1225:
1221:
1216:
1213:
1210:
1206:
1196:
1191:
1186:
1182:
1176:
1172:
1162:
1161:
1160:
1132:
1129:
1126:
1111:
1107:
1103:
1096:
1093:
1090:
1086:
1082:
1077:
1074:
1071:
1067:
1056:
1053:
1050:
1047:
1032:
1028:
1024:
1017:
1014:
1011:
1007:
1003:
998:
995:
992:
988:
978:
973:
968:
964:
954:
953:
952:
936:
930:
927:
924:
920:
916:
911:
908:
905:
901:
894:
889:
883:
880:
877:
873:
869:
864:
861:
858:
854:
847:
842:
838:
826:
798:
795:
792:
778:
774:
767:
764:
761:
757:
753:
748:
745:
742:
738:
727:
724:
721:
718:
704:
700:
693:
690:
687:
683:
679:
674:
671:
668:
664:
654:
649:
644:
640:
630:
629:
628:
612:
609:
606:
602:
598:
593:
590:
587:
583:
579:
574:
571:
568:
564:
560:
557:
552:
548:
544:
539:
536:
533:
529:
525:
520:
517:
514:
510:
506:
501:
497:
485:
479:
477:
476:
471:
466:
465:
454:
450:
447:
443:
442:
441:
434:
428:
424:
420:
416:
410:
405:
400:
396:
392:
390:
386:
382:
380:
376:
372:
371:
370:
368:
364:
360:
355:
349:
345:
341:
337:
334:
330:
326:
325:
324:
322:
318:
314:
313:217.5.143.100
310:
300:
298:
296:
292:
288:
284:
273:
271:
269:
265:
260:
254:
252:
248:
244:
240:
231:
228:
223:
220:
219:
218:
217:
214:
210:
206:
205:Lucas numbers
201:
200:
197:
193:
189:
184:
183:
180:
176:
168:
153:
149:
143:
140:
139:
136:
119:
115:
111:
110:
102:
96:
91:
89:
86:
82:
81:
77:
71:
68:
65:
61:
56:
52:
46:
38:
34:
29:
28:
19:
1991:
1629:
1606:— Preceding
1604:
1352:
1158:
827:
824:
486:
483:
461:
458:
433:source check
412:
406:
403:
359:Lucas number
356:
353:
307:— Preceding
304:
287:96.234.78.93
281:— Preceding
277:
259:User:Xinbone
255:
234:
221:
213:DavidCBryant
208:
204:
202:
188:Lucas number
185:
172:
148:Mid-priority
147:
107:
73:Mid‑priority
51:WikiProjects
336:PrimeHunter
262:—Preceding
237:—Preceding
179:PrimeHunter
123:Mathematics
114:mathematics
70:Mathematics
2027:Categories
470:Report bug
828:and item
487:The item
453:this tool
446:this tool
227:JRSpriggs
196:Gandalf61
1608:unsigned
459:Cheers.—
309:unsigned
283:unsigned
239:unsigned
363:my edit
264:undated
150:on the
41:C-class
1994:Ivigan
1576:=: -->
1475:=: -->
1324:=: -->
1257:=: -->
1130:=: -->
1051:=: -->
796:=: -->
722:=: -->
47:scale.
301:Links
190:into
2014:talk
1998:talk
1616:talk
340:talk
331:and
317:talk
291:talk
247:talk
2010:JBL
427:RfC
397:to
387:to
377:to
142:Mid
2029::
2016:)
2000:)
1992:--
1946:−
1922:−
1916:−
1864:−
1840:−
1769:−
1745:−
1739:−
1687:−
1663:−
1618:)
1555:−
1518:−
1507:−
1454:−
1417:−
1376::=
1300:−
1289:−
1233:−
1192::=
1094:−
1083:−
1015:−
974::=
937:10
909:−
862:−
754:−
746:−
672:−
650::=
610:−
599:−
572:−
518:−
440:.
435:}}
431:{{
342:)
319:)
293:)
249:)
2012:(
1996:(
1975:k
1971:F
1965:n
1961:F
1957:5
1954:=
1949:k
1943:n
1939:L
1933:k
1929:)
1925:1
1919:(
1911:k
1908:+
1905:n
1901:L
1895:,
1890:n
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1880:k
1876:L
1872:=
1867:k
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1857:L
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1837:(
1834:+
1829:k
1826:+
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1777:=
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1748:1
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1731:+
1728:n
1724:F
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979:{
969:n
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931:3
928:+
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912:3
906:n
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895:=
890:5
884:1
881:+
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874:L
870:+
865:1
859:n
855:L
848:=
843:n
839:F
799:2
793:m
779:m
775:F
768:m
765:+
762:n
758:F
749:m
743:n
739:F
728:;
725:1
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705:m
701:F
694:m
691:+
688:n
684:F
680:+
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669:n
665:F
655:{
645:n
641:L
613:2
607:n
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594:2
591:+
588:n
584:F
580:=
575:1
569:n
565:F
561:2
558:+
553:n
549:F
545:=
540:1
537:+
534:n
530:F
526:+
521:1
515:n
511:F
507:=
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468:(
455:.
448:.
338:(
315:(
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154:.
53::
20:)
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