A122909 a(n) = F(n+1)*F(2n+2) + F(n)*F(2n).

1, 4, 19, 79, 338, 1427, 6053, 25628, 108583, 459931, 1948354, 8253271, 34961561, 148099316, 627359147, 2657535383, 11257501522, 47687540107, 202007664157, 855718193164, 3624880442591, 15355239954179, 65045840274434

Offset: 0

Author

Paul Barry, Sep 18 2006

Keywords

• easy
• nonn

Comments

Let M be the matrix M(n,k)=F(k+1)*sum{j=0..n, (-1)^(n-j)C(n,j)C(j+1,k+1)}. a(n) gives the row sums of M^3.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Crossrefs

Cf. A000045, A037451, A061084, A163063.

Programs

Mathematica

Table[Fibonacci[n+1]Fibonacci[2n+2]+Fibonacci[n]Fibonacci[2n],{n,0,30}] (* or *) LinearRecurrence[{3,6,-3,-1},{1,4,19,79},30] (* Harvey P. Dale, Dec 11 2016 *)

Formula

G.f.: (1+x)*(1+x^2) / ( (x^2+4*x-1)*(x^2-x-1) ).

a(n) = (sqrt(5)+2)^n(sqrt(5)/5+3/5)-2^(-n-1)(sqrt(5)-1)^n(sqrt(5)/5+1/5)+ 2^(-n-1)(sqrt(5)+1)^n(sqrt(5)/5-1/5)(-1)^n+(sqrt(5)-2)^n(3/5-sqrt(5)/5)(-1)^n;

a(n) = (2*A163063(n) -A061084(n))/5. - R. J. Mathar, Jun 08 2016