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A274075
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Sum of n-th powers of the roots of x^3 + x^2 - 9*x - 1.
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7
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3, -1, 19, -25, 195, -401, 2131, -5545, 24323, -72097, 285459, -910009, 3407043, -11311665, 41065043, -139462985, 497736707, -1711838529, 6052005907, -20960815961, 73717030595, -256312368337, 898804827731, -3131899112169, 10964830193411, -38253117375201
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OFFSET
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0,1
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COMMENTS
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a(n) is always an integer.
a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial
x^3 + x^2 - 9*x - 1.
x1 = tan(Pi/7)/tan(4*Pi/7),
x2 = tan(4*Pi/7)/tan(2*Pi/7),
x3 = tan(2*Pi/7)/tan(Pi/7).
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LINKS
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FORMULA
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a(n) = (tan(Pi/7)/tan(4*Pi/7))^n + (tan(4*Pi/7)/tan(2*Pi/7))^n + (tan(2*Pi/7)/tan(Pi/7))^n.
a(n) = -a(n-1) + 9*a(n-2) + a(n-3) for n>2.
G.f.: (3+2*x-9*x^2) / (1+x-9*x^2-x^3). - Colin Barker, Jun 11 2016
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MATHEMATICA
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FullSimplify[Table[(Tan[Pi/7]/Tan[4*Pi/7])^n + (Tan[4*Pi/7]/Tan[2*Pi/7])^n + (Tan[2*Pi/7]/Tan[Pi/7])^n, {n, 0, 12}]] (* Wesley Ivan Hurt, Jun 11 2016 *)
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PROG
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(PARI) Vec((3+2*x-9*x^2)/(1+x-9*x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 11 2016
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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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