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A117152 Sum of product of Fibonacci and triangular numbers. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 25 08:33 EST 2021. Contains 341606 sequences. (Running on oeis4.)