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

Contribution from Giorgio Balzarotti (greenblue(AT)tiscali.it), Mar 11 2009: (Start)

Determinant of power series with alternate signs of gamma matrix with determinant 1!

a(n) = Determinant( A-A^2+ A^3-A^4+ A^5-... A^n)

where A is the submatrix A(1..2,1..2)= of the matrix with factorial determinant

A= [[1,1,1,1,1,1,...],[1,2,1,2,1,2,...],[1,2,3,1,2,3,...],

[1,2,3,4,1,2,...],[1,2,3,4,5,1,...],[1,2,3,4,5,6,...],...]

note: Determinant A(1..n,1..n)= (n-1)!

a(n) is even with respect to signs of power of A.

See A158039...A158050 for sequence with matrix 2!, 3!... (End)

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 27 2009: (Start)

Equals the INVERT transform of (1, 3, 2, 2, 2,...). Example: a(7) = 169

= (1, 1, 4, 9, 25, 64) dot (2, 2, 2, 2, 3, 1) = (2 + 2 + 8 + 18 + 75 + 64) = 169. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

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

f[n_]:=Fibonacci[n]^2; Array[f, 4!, 0] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 25 2009]

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

(Other) sage: [(fibonacci(n))^2 for n in xrange(0, 28)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]

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.

Sequence in context: A030481 A160190 A032127 this_sequence A121648 A133022 A028400

Adjacent sequences: A007595 A007596 A007597 this_sequence A007599 A007600 A007601

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)