A254399 A Fibonacci sum: a(n) = Sum_{j=0..n-1} F(j)^2*F(2*n-j), in which the F's are the Fibonacci numbers.

0, 2, 8, 41, 179, 794, 3422, 14706, 62754, 267179, 1135079, 4817276, 20429252, 86600786, 367005092, 1555075557, 6588493275, 27912159494, 118245265874, 500914535330, 2121959178350, 8988897300407, 38077930682063, 161301621015576, 683287035188904

Offset

1,2

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Curtis Greene and Herbert S. Wilf, Closed form summation of C-finite sequences, arXiv:math/0405574 [math.CO], 2005.

Eric Weisstein's MathWorld, Fibonacci Number

Wikipedia, Fibonacci number

Index entries for linear recurrences with constant coefficients, signature (6,-4,-18,14,0,-1).

Formula

a(n) = (1/2)*(F(2*n) + F(n)^2*F(n+1) - F(n)*F(n+1)^2 + F(n+1)^3 - F(2*n+1)).

G.f.: -(x^5+x^4-4*x^3+2*x^2)/(-x^6+14*x^4-18*x^3-4*x^2+6*x-1). - Alois P. Heinz, Jan 30 2015

a(n) = 3*(-1)^n*F(n-1)/10 + (-1)^n*F(n)/10 - F(2*n-1)/2 + F(3*n+1)/5. - Ehren Metcalfe, Mar 25 2016

Mathematica

F = Fibonacci; a[n_] := (1/2)*(F[2*n] + F[n]^2*F[n+1] - F[n]*F[n+1]^2 + F[n+1]^3 - F[2*n+1]); Array[a, 30]

Prog

(Magma) [(1/2)*(Fibonacci(2*n) + Fibonacci(n)^2*Fibonacci(n+1) - Fibonacci(n)*Fibonacci(n+1)^2 + Fibonacci(n+1)^3 - Fibonacci(2*n+1)): n in [1..25]]; // Vincenzo Librandi, Jan 30 2015
(PARI) concat(0, Vec(x^2*(1-x)*(2-2*x-x^2) / ((1-3*x+x^2)*(1+x-x^2)*(1-4*x-x^2)) + O(x^30))) \\ Colin Barker, Mar 26 2016

Crossrefs

Cf. A000045.

Sequence in context: A012597 A012534 A092354 * A337753 A340992 A348474

Adjacent sequences: A254396 A254397 A254398 * A254400 A254401 A254402

Keyword

easy,nonn

Author

Jean-François Alcover, Jan 30 2015

Status

approved