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%I #20 Feb 19 2016 06:43:14
%S 1,1,2,3,7,16,51,179,842,4601,33229,286284,3243665,44468561,800242506,
%T 17564890003,505712818663,17842259251624,825465630656435,
%U 46929863536852851,3498201665311407586,320978728492120944601
%N a(n+2) = a(n+1) + F(n+1)*a(n), where F = Fibonacci number (A000045) and a(0) = a(1) = 1.
%F Identity: a(n)*a(n+1)*a(n+4) + a(n)*a(n+2)^2 + a(n+1)^2*a(n+2) -
%F a(n)*a(n+1)*a(n+3) - a(n)*a(n+2)*a(n+3) - a(n+1)*a(n+2)^2 = 0. [_Emanuele Munarini_, Feb 18 2016]
%F a(n) = det(M(n)), where M(n) is the n x n tridiagonal matrix whose entries m(i,j) are defined as follows: m(i,i) = 1, m(i,i-1) = -1, m(i,i+1) = Fibonacci(i) = A000045(i) and m(i,j) = 0 otherwise (for i, j = 1..n). [_Emanuele Munarini_, Feb 19 2016]
%F a(n) ~ c * ((1 + sqrt(5))/2)^(n^2/4) / 5^(n/4), where c = 14.10659519071239329808481379222469071706794062942996705053477138... if n is even and c = 13.89554381027685566110211168629044351418320849411699988381803439... if n is odd. - _Vaclav Kotesovec_, Feb 19 2016
%t z[n_] := z[n] = z[n - 1] + Fibonacci[n - 1]z[n - 2] z[0] = 1 z[1] = 1
%o (Maxima) a[0]: 1$
%o a[1]: 1$
%o a[n] := a[n - 1] + fib(n - 1)*a[n - 2]$
%o makelist(a[n],n,0,25);
%o /* _Emanuele Munarini_, Feb 17 2016 */
%Y Cf. A000045, A269068.
%K nonn
%O 0,3
%A _Emanuele Munarini_, Dec 05 2003
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