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%I #38 Sep 08 2022 08:46:12
%S 6,42,110,210,342,506,702,930,1190,1482,1806,2162,2550,2970,3422,3906,
%T 4422,4970,5550,6162,6806,7482,8190,8930,9702,10506,11342,12210,13110,
%U 14042,15006,16002,17030,18090,19182,20306,21462,22650,23870,25122,26406
%N a(n) = (4*n+3)*(4*n+2).
%C Since 0 = Sin(Pi) = Sum_{n>=0}(-1)^n*Pi^(2n+1)/(2n+1)!, we can move the negative terms to the other side of the equation to get: Sum_{n>=0} Pi^(4n+1)/(4n+1)! = Sum_{n>=0}Pi^(4n+3)/(4n+3)!.
%C Now, if we let f(n) = Pi^(4n+1)/(4n+1)!, then the previous equation can be written as Sum_{n>=0}f(n) = Sum_{n>=0}(Pi^2/((4*n+3)*(4*n+2)))*f(n); a(n) is the n-th denominator on the right hand side.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 16*n^2 + 20*n + 6.
%F a(n) = 2*A033567(n+1).
%F G.f.: (6+24*x+2*x^2)/(1-x)^3. - _Vincenzo Librandi_, Apr 12 2015
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - _Vincenzo Librandi_, Apr 12 2015
%F a(n) = A016825(n)*A004767(n). - _Tom Edgar_, Apr 12 2015
%F a(n) = A002378(4*n+2) = 2*A000217(4*n+2). - _Ivan N. Ianakiev_, Apr 17 2015
%F E.g.f.: 2*exp(x)*(3+18*x+8*x^2). - _Wesley Ivan Hurt_, Apr 29 2020
%F From _Amiram Eldar_, Jan 03 2022: (Start)
%F Sum_{n>=0} 1/a(n) = Pi/8 - log(2)/4.
%F Sum_{n>=0} (-1)^n/a(n) = sqrt(2)*log(sqrt(2)+1)/4 - (sqrt(2)-1)*Pi/8. (End)
%t CoefficientList[Series[(6 + 24 x + 2 x^2) / (1 - x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Apr 12 2015 *)
%o (Magma) [16*n^2 + 20*n + 6: n in [0..40]]; // _Vincenzo Librandi_, Apr 12 2015
%o (PARI) vector(50,n,(4*n-1)*(4*n-2)) \\ _Derek Orr_, Apr 13 2015
%Y Cf. A000217, A002378, A004767, A016825, A033567.
%K nonn,easy
%O 0,1
%A _Bruce Zimov_, Apr 10 2015
%E More terms from _Vincenzo Librandi_, Apr 12 2015
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