This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363753 #60 Aug 01 2023 22:01:25
%S A363753 0,0,1,-2,13,-47,213,-879,3762,-15873,67342,-285098,1207966,-5116586,
%T A363753 21674919,-91815276,388937619,-1647563169,6979194475,-29564334305,
%U A363753 125236542640,-530510487155,2247278519916,-9519624520452,40325776676748,-170822731106052,723616701297373
%N 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).
%C A363753 Alternating sum of the product of three consecutive Fibonacci numbers, divided by two.
%C A363753 Can also be seen as the alternating sum of the Fibonomial coefficients (n+1,3), A001655.
%C A363753 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.
%H A363753 K. Subba Rao, <a href="https://www.jstor.org/stable/2307147">Some properties of Fibonacci numbers</a>, The American Mathematical Monthly, 60(10):680-684, December 1953.
%H A363753 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-2,9,-3,-4,1).
%F A363753 a(n) = ((-1)^n*(F(n+1)^3 - F(n)^3) + F(n+2) - 2)/8.
%F A363753 a(n) = ((-1)^n*F(3*n+1) + 4*F(n+2) - 5)/20.
%F A363753 a(n) = -2*a(n-1) + 9*a(n-2) - 3*a(n-3) - 4*a(n-4) + a(n-5).
%F A363753 a(-n) = A215037(n-3).
%F A363753 G.f.: x^2/((1 - x)*(1 + 4*x - x^2)*(1 - x - x^2)).
%F A363753 20*a(n) = (-1)^n*A033887(n) + 4*A000045(n+2) - 5. - _R. J. Mathar_, Jun 27 2023
%t A363753 LinearRecurrence[{-2, 9, -3, -4, 1}, {0, 0, 1, -2, 13}, 27]
%Y A363753 Cf. A000045, A001655.
%Y A363753 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).
%Y A363753 A005968: F(k)^3, A119284: (-1)^k*F(k)^3, A215037: F(k-1)*F(k)*F(k+1),
%Y A363753 A363753: (-1)^k*F(k-1)*F(k)*F(k+1), A163198: F(2k)^3, A163200: F(2k+1)^3,
%Y A363753 A256178: F(2k)*F(2k+1)*F(2k+2), this sequence: (-1)^k*F(k-1)*F(k)*F(k+1),
%Y A363753 A363754: F(2k-1)*F(2k)*F(2k+1).
%K A363753 sign,easy
%O A363753 0,4
%A A363753 _Hans J. H. Tuenter_, Jun 19 2023