縞吹计

猭瓣计厩產縞吹 (Edouard Lucas 1842-1891)

(酚The MacTutor History of Mathematics Achievehttp://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 = 0ㄤ 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......

иΤ㎝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 1118 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-2ㄤいぃ琌 L1 = 1 L2 = 3

┮縞吹计Τ1, 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

иΤ㎝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.