Hi,
Here is the OEIS sequence description: A136429 a(n) = sum( F(k+1)^2 F(n-k+1)^2, k = 0..n ) where F(n) = Fibonacci number (A000045). 1, 2, 9, 26, 84, 250, 747, 2182, 6323, 18132, 51624, 146004, 410677, 1149578, 3204477, 8899502, 24634620, 67990414, 187154271, 513939214, 1408246247, 3851081256, 10512259920, 28647203880, 77946605545, 211782868754 OFFSET
0,2 FORMULA
G.f.: (1-x)^2/((1+x)^2(1-3x+x^2)^2).
Recurrence: a(n+6) = 4 a(n+5) - 10 a(n+3) + 4 a(n+1) - a(n).
AUTHOR
Emanuele Munarini (emanuele.munarini(AT)polimi.it), Apr 01 2008
So I tried PURRS http://www.cs.unipr.it/purrs/ PURRS Demo Results Exact solution for x(n) = -x(6+n)+4*x(5+n)-10*x(3+n)+4*x(1+n) for the initial conditions x(0) = 1 x(1) = 2 x(2) = 9 x(3) = 26 x(4) = 84 x(5) = 250 x(n) = -(-1)^n*n-2/5*(3/2+1/2*sqrt(5))^n+9/5*(-1)^n+4/5*(3/2+1/2*sqrt(5))^n*sqrt(5)-4/5 *(3/2-1/2*sqrt(5))^n*sqrt(5)-2/5*(3/2-1/2*sqrt(5))^n for each n >= 0 Then I have defined sequence in PARI using close formula generated by PURRS (21:16) gp > a(n)=-(-1)^n*n-2/5*(3/2+1/2*sqrt(5))^n+9/5*(-1)^n+4/5*(3/2+1/2*sqrt(5))^n*sqrt(5)-4/5 *(3/2-1/2*sqrt(5))^n*sqrt(5)-2/5*(3/2-1/2*sqrt(5))^n
But it doesn't even give initial conditions ... ? (21:16) gp > a(0) %5 = 1.000000000000000000000000000 (21:16) gp > a(1) %6 = 2.000000000000000000000000000 (21:17) gp > a(3) %7 = 26.00000000000000000000000000 (21:17) gp > a(4) %8 = 63.00000000000000000000000000 (21:17) gp > a(5) %9 = 174.0000000000000000000000000 (21:19) gp > a(6) %10 = 443.0000000000000000000000000
Did I make a mistake in above or ... ?
Ciao, Regards, Alex