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%I #25 Mar 29 2016 03:19:02
%S 1,1,1,5,6,29,35,169,204,985,1189,5741,6930,33461,40391,195025,235416,
%T 1136689,1372105,6625109,7997214,38613965,46611179,225058681,
%U 271669860,1311738121,1583407981,7645370045,9228778026,44560482149,53789260175,259717522849
%N a(n) = product( 4 +4*cos(k*Pi/n)^2, k=1..(n-1)/2 ).
%C Sequence of products: Product[m + 4*Cos[k*Pi/n]^2, {k, 1, (n - 1)/2}; m=1,2,3,4-> A000045, A002530, A136211 and this one.
%C Apparently the same as A041011 after the initial term. - _R. J. Mathar_, Nov 27 2008
%H Colin Barker, <a href="/A152118/b152118.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F From _Colin Barker_, Oct 23 2013: (Start)
%F a(n) = 6*a(n-2)-a(n-4) for n>4.
%F G.f.: (x^4-x^3-5*x^2+x+1) / ((x^2-2*x-1)*(x^2+2*x-1)). (End)
%F a(n) = ((-(-1 - sqrt(2))^n - 3*(1-sqrt(2))^n + (-1+sqrt(2))^n + 3*(1+sqrt(2))^n)) / (8*sqrt(2)) for n>0. - _Colin Barker_, Mar 28 2016
%F E.g.f.: (1/(2*sqrt(2)))*(2*sqrt(2) + (2*cosh(x) + sinh(x))*sinh(sqrt(2)*x)). - _G. C. Greubel_, Mar 28 2016
%t a = Table[Product[4 + 4*Cos[k*Pi/n]^2, {k, 1, (n - 1)/2}], {n, 0, 30}]; FullSimplify[ExpandAll[a]] Round[%]
%t Join[{1}, LinearRecurrence[{0, 6, 0, -1}, {1, 1, 5, 6}, 20]] (* _G. C. Greubel_, Mar 28 2016 *)
%o (PARI) a(n) = round(prod(k=1, (n-1)/2, 4 + 4*cos(k*Pi/n)^2)) \\ _Colin Barker_, Oct 23 2013
%o (PARI) Vec((x^4-x^3-5*x^2+x+1)/((x^2-2*x-1)*(x^2+2*x-1)) + O(x^50)) \\ _Colin Barker_, Mar 28 2016
%Y Cf. A000045, A002530, A136211.
%K nonn,easy
%O 0,4
%A _Roger L. Bagula_ and _Gary W. Adamson_, Nov 24 2008
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