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A232719
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Sum_{k=1,...,2n} (-1)^k binomial(8*n,4*k).
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2
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-69, 9231, -1254465, 170459391, -23162405889, 3147359850495, -427670341173249, 58112808641953791, -7896499249846943745, 1072994093040913088511, -145800852665566628937729, 19811748057028406926114815, -2692064922113214275888611329, 365803841438484687010033303551
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OFFSET
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1,1
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COMMENTS
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All elements of this sequence are multiples of 3. Any proof?
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LINKS
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FORMULA
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a(n) = (-1)^n/2 * ((2+sqrt(2))^(4*n) + (2-sqrt(2))^(4*n)) - 1. - Vaclav Kotesovec, Dec 06 2013
G.f.: 3*x*(28*x+23) / ((x-1)*(16*x^2+136*x+1)). - Colin Barker, Dec 06 2013
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MAPLE
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MATHEMATICA
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A[n_] := Sum[(-1)^k Binomial[8 n, 4 k], {k, 1, 2n}]; Array[A, 33]
Table[FullSimplify[(-1)^n/2*((2+Sqrt[2])^(4*n)+(2-Sqrt[2])^(4*n))-1], {n, 1, 15}] (* Vaclav Kotesovec, Dec 06 2013 *)
CoefficientList[Series[3 (28 x + 23) / ((x - 1) (16 x^2 + 136 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 09 2014 *)
LinearRecurrence[{-135, 120, 16}, {-69, 9231, -1254465}, 20] (* Harvey P. Dale, Jan 31 2023 *)
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PROG
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(PARI) Vec(3*x*(28*x+23)/((x-1)*(16*x^2+136*x+1)) + O(x^100)) \\ Colin Barker, Nov 09 2014
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CROSSREFS
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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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