0,4

a(n)*(-1)^(n+1) = (2*(1-T(n,-3/2))/5), n>=0, with Chebyshev's polynomials T(n,x) of the first kind, is the r=-1 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 2004

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 8.

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 130.

R. P. Stanley, Enumerative Combinatorics I, Example 4.7.14, p. 251.

T. D. Noe, Table of n, a(n) for n=0..200

D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux,

T. Mansour, A note on sum of k-th power of Horadam's sequence

T. Mansour, Squaring the terms of an ell-th order linear recurrence

P. Stanica, Generating functions, weighted and non-weighted sums of powers...

a(0) = 0, a(1) = 1; a(n) = a(n-1) + Sum(a(n-i)) + k, 0 <= i < n where k = 1 when n is odd, or k = -1 when n is even. E.g. a(2) = 1 = 1 + (1 + 1 + 0) - 1, a(3) = 4 = 1 + (1 + 1 + 0) + 1, a(4) = 9 = 4 + (4 + 1 + 1 + 0) - 1, a(5) = 25 = 9 + (9 + 4 + 1 + 1 + 0) + 1. - Sadrul Habib Chowdhury (adil040(AT)yahoo.com), Mar 02 2004

G.f.: x(1-x)/((1+x)(1-3x+x^2)). a(n)=2a(n-1)+2a(n-2)-a(n-3), n>2. a(0)=0, a(1)=1, a(2)=1. a(-n)=a(n).

(1/5)[2*Fibonacci(2n+1) - Fibonacci(2n) - 2(-1)^n]. - R. Stephan, May 14 2004

a(n) = F(n-1)F(n+1) - (-1)^n = A059929(n-1) - A033999(n).

a(n) = right term of M^n * [1 0 0] where M = the 3X3 matrix [1 2 1 / 1 1 0 / 1 0 0]. M^n * [1 0 0] = [a(n+1) A001654(n) a(n)]. E.g. a(4) = 9 since M^4 * [1 0 0] = [25 15 9] = [a(5) A001654(4) a(4)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 19 2004

Sum_(j=0..2n) binomial(2n,j) a(j)= 5^(n-1) A005248(n+1) for n>=1 [P. Stanica]. sum_(j=0..2n+1) binomial(2n+1,j) a(j)=5^n A001519(n+1) [P. Stanica]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 16 2006

with (combinat):seq(mul(fibonacci(n), k=1..2), n=0..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 21 2007

(PARI) a(n)=fibonacci(n)^2

Cf. A001254, A079291.

Bisection of A006948 and A074677. First differences of A001654.

Equals A080097(n-2) + 1. Cf. A061646, A065885.

Cf. A056570.

Cf. A001654.

Second row of array A103323.

Adjacent sequences: A007595 A007596 A007597 this_sequence A007599 A007600 A007601

Sequence in context: A068888 A030481 A032127 this_sequence A121648 A133022 A028400

nonn,easy,nice

njas, Robert G. Wilson v (rgwv(AT)rgwv.com)