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

Author

Mitch Harris, Feb 28 2006

Keywords

• nonn
• easy

References

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

Crossrefs

Cf. A086926, A014286.

Programs

Mathematica

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

PARI

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

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).

a(n)-a(n-1) = A086926(n). - R. J. Mathar, May 16 2025