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A363753
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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).
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4
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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
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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FORMULA
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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).
G.f.: x^2/((1 - x)*(1 + 4*x - x^2)*(1 - x - x^2)).
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MATHEMATICA
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LinearRecurrence[{-2, 9, -3, -4, 1}, {0, 0, 1, -2, 13}, 27]
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CROSSREFS
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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).
A256178: F(2k)*F(2k+1)*F(2k+2), this sequence: (-1)^k*F(k-1)*F(k)*F(k+1),
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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