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%I #28 Jan 04 2023 03:59:24
%S 1,7,13,97,181,1351,2521,18817,35113,262087,489061,3650401,6811741,
%T 50843527,94875313,708158977,1321442641,9863382151,18405321661,
%U 137379191137,256353060613,1913445293767,3570537526921,26650854921601,49731172316281,371198523608647
%N a(2*n) = A001570(n), a(2*n+1) = A011943(n+1).
%C See also A110294 (compare program code).
%C a(2*n+1) = (a(2*n) + a(2*n+2))/2 and see A232765 for Diophantine equation that produces a sequence related to a(n). - _Richard R. Forberg_, Nov 30 2013
%H Colin Barker, <a href="/A110293/b110293.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 (0,14,0,-1).
%F G.f.: (1+7*x-x^2-x^3) / ((1-4*x+x^2)*(1+4*x+x^2)).
%F From _Colin Barker_, Nov 01 2016: (Start)
%F a(n) = (3-(-1)^n)*((-3+2*sqrt(3))*(2-sqrt(3))^n + (3+2*sqrt(3))*(2+sqrt(3))^n )/(8*sqrt(3)).
%F a(n) = 14*a(n-2) - a(n-4) for n>3. (End)
%F a(n) = (1/4)*(3 - (-1)^n)*(2*A001353(n) - A001353(n-1)). - _G. C. Greubel_, Jan 04 2023
%p seriestolist(series((1+7*x-x^2-x^3)/((1-4*x+x^2)*(1+4*x+x^2)), x=0, 25));
%t CoefficientList[Series[(1+7x-x^2-x^3)/((1-4x+x^2)(1+4x+x^2)), {x, 0, 25}], x] (* _Michael De Vlieger_, Nov 01 2016 *)
%o (PARI) Vec((1+7*x-x^2-x^3)/((1-4*x+x^2)*(1+4*x+x^2)) + O(x^30)) \\ _Colin Barker_, Nov 01 2016
%o (Magma)
%o A001353:= func< n | Evaluate(ChebyshevSecond(n+1), 2) >;
%o [(3-(-1)^n)*(2*A001353(n) - A001353(n-1))/4: n in [0..40]]; // _G. C. Greubel_, Jan 04 2023
%o (SageMath)
%o def A001353(n): return chebyshev_U(n,2)
%o [(3-(-1)^n)*(2*A001353(n) - A001353(n-1))/4 for n in range(41)] # _G. C. Greubel_, Jan 04 2023
%Y Cf. A001353, A001570, A011943, A110294, A232765.
%K easy,nonn
%O 0,2
%A _Creighton Dement_, Jul 18 2005
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