A117152 Sum of product of Fibonacci and triangular numbers.

0, 0, 1, 7, 25, 75, 195, 468, 1056, 2280, 4755, 9650, 19154, 37328, 71635, 135685, 254125, 471317, 866669, 1581620, 2866970, 5165630, 9256871, 16507092, 29304660, 51812160, 91264885, 160207603, 280340161, 489117135, 851054535

Offset

0,4

References

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003

Links

Table of n, a(n) for n=0..30.

Formula

a(n) = Sum_{k=2..n} C(k,2)*F(k), where F(n) = A000045(n), the Fibonacci numbers and C(n, 2) = A000217(n-1), the triangular numbers, n(n-1)/2.

a(n) = C(n,2) F(n+2) - n F(n+3) + F(n+5) - 5.

G.f.: x^2(1 + 3x + x^3)/((1 - x)(1 - x - x^2)^3).

Mathematica

Binomial[n, 2]Fibonacci[n + 2] - n Fibonacci[n + 3] + Fibonacci[n + 5] - 5

Prog

(PARI) a(n) = sum(k=2, n, k*(k-1)*fibonacci(k)/2); \\ Michel Marcus, Feb 28 2019

Crossrefs

Cf. A086926, A014286.

Sequence in context: A299269 A048477 A294837 * A155281 A155254 A155295

Adjacent sequences: A117149 A117150 A117151 * A117153 A117154 A117155

Keyword

nonn,easy

Author

Mitch Harris, Feb 28 2006

Status

approved