A215038 Partial sums of A066259: a(n) = sum(F(k+1)^2*F(k),k=0..n), n>=0, with the Fibonacci numbers F=A000045.

0, 1, 5, 23, 98, 418, 1770, 7503, 31779, 134629, 570284, 2415788, 10233404, 43349461, 183631161, 777874251, 3295127934, 13958386366, 59128672790, 250473078515, 1061020985255, 4494557022121, 19039249069560, 80651553307128

Offset

0,3

Comments

For a derivation of the explicit form of this sum see the link under A215308 on the partial summation formula, eq. (7).

Links

Table of n, a(n) for n=0..23.

Index entries for linear recurrences with constant coefficients, signature (4, 3, -9, 2, 1).

Formula

a(n) = sum(A066259(k),k=0..n) = sum(F(k+1)^2*F(k),k=0..n), n >= 0, with A066259(0)=0.

a(n) = (F(n+2)*F(n+1)^2 - (-1)^n*(F(n) + (-1)^n)/2 = (A066258(n+1) - (-1)^n*A008346(n))/2, n >= 0.

O.g.f.: x*(1+x)/((1+x-x^2)*(1-4*x-x^2)*(1-x)) (from A066259).

Example

a(2) = 0 + 1^2*1 + 2^2*1 = 1 + 4 = 5.

Crossrefs

Cf. A001655, A215037.

Sequence in context: A323922 A368835 A119012 * A084615 A181331 A268400

Adjacent sequences: A215035 A215036 A215037 * A215039 A215040 A215041

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 09 2012

Status

approved