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A275546
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a(n) = (tan(1*Pi/11))^(2*n) + (tan(2*Pi/11))^(2*n) + (tan(3*Pi/11))^(2*n) + (tan(4*Pi/11))^(2*n) + (tan(5*Pi/11))^(2*n).
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5
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5, 55, 2365, 113311, 5476405, 264893255, 12813875437, 619859803695, 29985188632421, 1450508002869079, 70167091762786205, 3394273427239643839, 164195092176119969173, 7942798031108524622951, 384226104001681151724877, 18586611219134532494467151, 899111520569015285343455941, 43493755633501102693569684087, 2103973462501643822799172235773
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OFFSET
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0,1
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COMMENTS
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(tan(1*Pi/11))^(2*n), (tan(2*Pi/11))^(2*n), (tan(3*Pi/11))^(2*n),(tan(4*Pi/11))^(2*n), (tan(5*Pi/11))^(2*n) are roots of the polynomial x^5 - 55x^4 + 330x^3 - 462x^2 + 165x - 11.
Sum_{k=1..(m-1)/2)} tan^(2n) (k*Pi/m) is an integer when m >= 3 is an odd integer (see AMM link); this sequence is the particular case m = 11. All terms are odd. - Bernard Schott, Apr 24 2022
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LINKS
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FORMULA
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a(-2) = 141, a(-1) = 15, a(0) = 5, a(1) = 55, a(2) = 2365.
a(n) = +55*a(n-1)-330*a(n-2)+462*a(n-3)-165*a(n-4)-11*a(n-5) for n > 2.
a(n) ~ k^n where k = 48.37415... is the largest real root of x^5 - 55x^4 + 330x^3 - 462x^2 + 165x - 11. - Charles R Greathouse IV, Aug 01 2016
G.f.: (5-220*x+990*x^2-924*x^3+165*x^4) / (1-55*x+330*x^2-462*x^3+165*x^4-11*x^5). - Colin Barker, Aug 02 2016
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PROG
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(PARI) a(n)=([0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; 11, -165, 462, -330, 55]^n*[5; 55; 2365; 113311; 5476405])[1, 1] \\ Charles R Greathouse IV, Aug 01 2016
(PARI) Vec((5-220*x+990*x^2-924*x^3+165*x^4)/(1-55*x+330*x^2-462*x^3+165*x^4-11*x^5) + O(x^20)) \\ Colin Barker, Aug 02 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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