%I A007598 M3364
%S A007598 0,1,1,4,9,25,64,169,441,1156,3025,7921,20736,54289,142129,
%T A007598 372100,974169,2550409,6677056,17480761,45765225,119814916,
%U A007598 313679521,821223649,2149991424,5628750625,14736260449,38580030724
%N A007598 F(n)^2, where F() = Fibonacci numbers A000045.
%C A007598 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
%C A007598 Contribution from Giorgio Balzarotti (greenblue(AT)tiscali.it), Mar 11 
               2009: (Start)
%C A007598 Determinant of power series with alternate signs of gamma matrix with 
               determinant 1!
%C A007598 a(n) = Determinant( A-A^2+ A^3-A^4+ A^5-... A^n)
%C A007598 where A is the submatrix A(1..2,1..2)= of the matrix with factorial determinant
%C A007598 A= [[1,1,1,1,1,1,...],[1,2,1,2,1,2,...],[1,2,3,1,2,3,...],
%C A007598 [1,2,3,4,1,2,...],[1,2,3,4,5,1,...],[1,2,3,4,5,6,...],...]
%C A007598 note: Determinant A(1..n,1..n)= (n-1)!
%C A007598 a(n) is even with respect to signs of power of A.
%C A007598 See A158039...A158050 for sequence with matrix 2!, 3!... (End)
%C A007598 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 27 2009: 
               (Start)
%C A007598 Equals the INVERT transform of (1, 3, 2, 2, 2,...). Example: a(7) = 169
%C A007598 = (1, 1, 4, 9, 25, 64) dot (2, 2, 2, 2, 3, 1) = (2 + 2 + 8 + 18 + 75 
               + 64) = 169. (End)
%D A007598 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 8.
%D A007598 R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 130.
%D A007598 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007598 R. P. Stanley, Enumerative Combinatorics I, Example 4.7.14, p. 251.
%H A007598 T. D. Noe, <a href="b007598.txt">Table of n, a(n) for n=0..200</a>
%H A007598 D. Foata and G.-N. Han, <a href="http://www-irma.u-strasbg.fr/~foata/
               paper/pub71.html">Nombres de Fibonacci et polynomes orthogonaux</
               a>,
%H A007598 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0302015">A note on 
               sum of k-th power of Horadam's sequence</a>
%H A007598 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0303138">Squaring the 
               terms of an ell-th order linear recurrence</a>
%H A007598 P. Stanica, <a href="http://arXiv.org/abs/math.CO/0010149">Generating 
               functions, weighted and non-weighted sums of powers...</a>
%H A007598 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A007598 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A007598 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
%F A007598 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).
%F A007598 (1/5)[2*Fibonacci(2n+1) - Fibonacci(2n) - 2(-1)^n]. - R. Stephan, May 
               14 2004
%F A007598 a(n) = F(n-1)F(n+1) - (-1)^n = A059929(n-1) - A033999(n).
%F A007598 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
%F A007598 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
%p A007598 with (combinat):seq(mul(fibonacci(n), k=1..2), n=0..27); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Sep 21 2007
%t A007598 f[n_]:=Fibonacci[n]^2;Array[f,4!,0] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Oct 25 2009]
%o A007598 (PARI) a(n)=fibonacci(n)^2
%o A007598 (Other) sage: [(fibonacci(n))^2 for n in xrange(0, 28)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
%Y A007598 Cf. A001254, A079291.
%Y A007598 Bisection of A006948 and A074677. First differences of A001654.
%Y A007598 Equals A080097(n-2) + 1. Cf. A061646, A065885.
%Y A007598 Cf. A056570.
%Y A007598 Cf. A001654.
%Y A007598 Second row of array A103323.
%Y A007598 Sequence in context: A030481 A160190 A032127 this_sequence A121648 A133022 
               A028400
%Y A007598 Adjacent sequences: A007595 A007596 A007597 this_sequence A007599 A007600 
               A007601
%K A007598 nonn,easy,nice
%O A007598 0,4
%A A007598 N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)