a(n) = (1/2)*Sum_{k=0..n} binomial(n, k)*F(3*k) where F(k) denotes the k-th Fibonacci number.
a(n) = sqrt(5)*((3+sqrt(5))^n - (3-sqrt(5))^n)/10. -
Paul Barry, Aug 25 2003
From Mario Catalani (mario.catalani(AT)unito.it), Jul 22 2004: (Start)
a(n) = Sum_{0..floor((n-1)/2)} C(n, 2k+1)*5^k*3^(n-2k-1).
a(n) = 2^(n-1)*F(2n). (End)
a(n) is the rightmost term in M^n * [1 0] where M is the 2X2 matrix [5 1 / 1 1]. The characteristic polynomial of M = x^2 - 6x + 4. a(n)/a(n-1) tends to (3 + sqrt(5)), a root of the polynomial and an eigenvalue of M. -
Gary W. Adamson, Dec 16 2004
a(n) = Sum_{k = 0..n} Sum_{j = 0..n} C(n, j)*C(j, k)*F(j+k)/2. -
Paul Barry, Feb 14 2005
If p[i] = (4^i-1)/3, and if A is the Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i = j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = det A. -
Milan Janjic, May 08 2010
a(n) = 5a(n - 1) + a(n - 2) + a(n - 3) + ... + a(1) + 1. -
Gary W. Adamson, Feb 18 2011
E.g.f.: exp(3*x) * sinh(sqrt(5)*x) / sqrt(5). -
Amiram Eldar, Jan 04 2026
