A363753 a(n) = Sum_{k=0..n} (-1)^k*F(k-1)*F(k)*F(k+1)/2, where F(n) is the Fibonacci number A000045(n).

0, 0, 1, -2, 13, -47, 213, -879, 3762, -15873, 67342, -285098, 1207966, -5116586, 21674919, -91815276, 388937619, -1647563169, 6979194475, -29564334305, 125236542640, -530510487155, 2247278519916, -9519624520452, 40325776676748, -170822731106052, 723616701297373

Offset: 0

Author

Hans J. H. Tuenter, Jun 19 2023

Keywords

• sign
• easy

Comments

Alternating sum of the product of three consecutive Fibonacci numbers, divided by two.

Can also be seen as the alternating sum of the Fibonomial coefficients (n+1,3), A001655.

This sequence is part of a suite of sums over triple products of Fibonacci numbers. Subba Rao (1953) gives closed-form expressions for several Fibonacci sums of this type.

Links

• K. Subba Rao, Some properties of Fibonacci numbers, The American Mathematical Monthly, 60(10):680-684, December 1953.
• Index entries for linear recurrences with constant coefficients, signature (-2,9,-3,-4,1).

Crossrefs

Cf. A000045, A001655.

Other sequences with the product of three Fibonacci numbers as a summand (the sequence may have a shifted [and scaled] version of the summand given here).

A005968: F(k)^3, A119284: (-1)^k*F(k)^3, A215037: F(k-1)*F(k)*F(k+1),

A363753: (-1)^k*F(k-1)*F(k)*F(k+1), A163198: F(2k)^3, A163200: F(2k+1)^3,

A256178: F(2k)*F(2k+1)*F(2k+2), this sequence: (-1)^k*F(k-1)*F(k)*F(k+1),

A363754: F(2k-1)*F(2k)*F(2k+1).

Programs

Mathematica

LinearRecurrence[{-2, 9, -3, -4, 1}, {0, 0, 1, -2, 13}, 27]

Formula

a(n) = ((-1)^n*(F(n+1)^3 - F(n)^3) + F(n+2) - 2)/8.

a(n) = ((-1)^n*F(3*n+1) + 4*F(n+2) - 5)/20.

a(n) = -2*a(n-1) + 9*a(n-2) - 3*a(n-3) - 4*a(n-4) + a(n-5).

a(-n) = A215037(n-3).

G.f.: x^2/((1 - x)*(1 + 4*x - x^2)*(1 - x - x^2)).

20*a(n) = (-1)^n*A033887(n) + 4*A000045(n+2) - 5. - R. J. Mathar, Jun 27 2023