縞吹计
猭瓣计厩產縞吹 (Edouard Lucas 1842-1891)
(酚The MacTutor History of Mathematics Achievehttp://www-gap.dcs.st-and.ac.uk/~history/ )
縞吹计
縞吹计 (Lucas Sequence) ㎝禣猧计 (Fibonacci Sequence) Τ馋闽玒珿セざ残禣猧计 (Fibonacci Number) 眔縞吹计睰彻
﹚竡俱计 P ㎝ Q ㄏ D = P2 - 4Q > 0
眖τ眔よ祘 x2 - Px + Q = 0ㄤ a, b
瞷﹚竡縞吹计
Un(P,Q) = (an - bn) / (a-b) の Vn(P,Q) = an + bn
ㄤい n 獶璽俱计眔 U0(P,Q) = 0 U1(P,Q) = 1 V0(P,Q) = 2 V1(P,Q) = P......
иΤ㎝縞吹计闽单Α
Um+n
= UmVn - anbnUm-n
Vm+n = VmVn - anbnVm-n
Um+1
= P*Um - Q*Um-1 Vm+1 = P*Vm
- Q*Vm-1 ( n = 1)
U2n
= UnVn V2n = Vn2
- Qn
U2n+1
= Un+1Vn - Qn V2n+1 = Vn+1Vn
- PQn
ㄌ妓脯莫计
璝 (P,Q) = (1,-1)и獽Τ Un 禣猧计
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4141 6765单
τ Vn 縞吹计 (Lucas Number)
2 1 3 4 7 1118 29 47 76 123 199 322 521 843 1364 2207 3571 5781 9349 单
璝 (P,Q) = (2,-1)и獽Τ Un ㄘ焊计 (Pell Number)
0 1 2 5 12 29 70 169 408 985 2378 5741单
τ Vn ㄘ焊 - 縞吹计 (Pell - Lucas Number) (冈ǎゅㄘ焊计)
2 2 6 14 34 82 198 478 1154 2786 6726单
单常琌计厩Τ计 (Sequence)
縞吹计单Α
縞吹计 (虏癘 Ln) Τ┦借㎝禣猧计 Ln = Ln-1 + Ln-2ㄤいぃ琌 L1 = 1 L2 = 3
┮縞吹计Τ1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...... (OEIS A000204)讽いキよ计 (Square Number) Τ 1 ㎝ 4硂琌パ (John H. E. Cohn) 靡τ计 (Prime Number)縞吹计 (Lucas Prime) 玥Τ 3, 7, 11, 29, 47, ...... 癸ヴ计 m Ln 穦琌 Lnm ┮т碝縞吹计穦σ納计计 p ┪ 2 よ经
иΤ㎝縞吹计闽单Α
Ln2
- Ln-1Ln+1 = 5 (-1)n
L12
+ L22 + ...... + Ln2 = LnLn+1
- 2
Lm+n
= (5FmFn + LmLn) / 2 (Αい Fn
禣猧计)
Lm-n
= (-1)n (LmLn - 5FmFn)
/ 2
Ln2
- 5Fn2 = 4 (-1)n
把σゅ膍の呼
Caldwell, C. K. "The Top Twenty: Lucas Number." http://primes.utm.edu/top20/page.php?id=48.
Ribenboim, P. "The Little Book of Bigger Prime" , New York: Springer-Verlag, 1991
Weisstein, E. W. "Lucas Number." From MathWorld. http://mathworld.wolfram.com/LucasNumber.html.
