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A215037 a(n) = sum(fibonomial(k+3,3), k=0..n), n>=0. 3
1, 4, 19, 79, 339, 1431, 6072, 25707, 108922, 461362, 1954426, 8278978, 35070483, 148560678, 629313573, 2665814361, 11292572005, 47836100785, 202636977730, 858384007525, 3636173014596, 15403076054964, 65248477252164 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sum is obtained from the m=2 member of the m-family of sums s(m;n) := sum(F(k+m)*F(k+1)*F(k),k=0..n), n>=0, given by

  (F(n+m)*F(n+2)*F(n+1) - (-1)^n*F(m)*A008346(n))/2 with A008346(n) = (F(n) + (-1)^n), where F = A000045.

The formula for s(m;n), m>=0, n>=0, follows by induction on m, using the sums for m=0 and m=1. s(0,n) = F(n+1)*(F(n+1)^2 - (-1)^n)/2 = F(n+2)*F(n+1)*F(n)/2 (see A001655(n-1)), and s(1,n) = (F(n+2)*F(n+1)^2 - (-1)^n*A008346(n))/2 (see A215038). For the formulas for s(0,n) and s(1,n) see also the link on partial summation, eqs. (6) and (7). There sum(fibonomial(k+2,k),k=0..n) is obtained more directly in eq. (5) with the help of the partial summation formula.

LINKS

Table of n, a(n) for n=0..22.

Wolfdieter Lang, Partial summation formula applied to sums over cubes of Fibonacci numbers.

FORMULA

a(n) = sum(F(k+3)F(k+2)*F(k+1)/2, k=0..n), n>=0, with the Fibonacci numbers F = A000045.

a(n) = (F(n+3)^2*F(n+2) + (-1)^n*A008346(n+1))/4, n>=0, with A008346(n) = F(n) + (-1)^n. See a comment above.

O.g.f.: 1/((1+x-x^2)*(1-4*x-x^2)*(1-x)) (from the one of the fibonomials A001655).

EXAMPLE

a(3) = (2*1*1/2  + 3*2*1/2 +   5*3*2/2 +8*5*3/2 = 1 + 3 + 15 + 60 = 79.

CROSSREFS

Cf, A215038.

Sequence in context: A156760 A320088 A122909 * A181300 A027240 A050914

Adjacent sequences:  A215034 A215035 A215036 * A215038 A215039 A215040

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 09 2012

STATUS

approved

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Last modified March 20 19:23 EDT 2019. Contains 321349 sequences. (Running on oeis4.)