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%I #37 May 04 2021 03:12:09
%S 0,14,44,90,152,230,324,434,560,702,860,1034,1224,1430,1652,1890,2144,
%T 2414,2700,3002,3320,3654,4004,4370,4752,5150,5564,5994,6440,6902,
%U 7380,7874,8384,8910,9452,10010,10584,11174,11780,12402,13040,13694,14364,15050
%N a(n) = 2*n*(4*n + 3).
%C The inverse binomial transform is [0, 14, 16, 0, 0, 0, ...]. - _R. J. Mathar_, May 06 2008
%C Sequence found by reading the line from 0, in the direction 0, 14, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the even hexagonal numbers A014635 in the same spiral. - _Omar E. Pol_, Sep 03 2011
%H G. C. Greubel, <a href="/A033587/b033587.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 2*A033954(n).
%F O.g.f.: 2*x*(7+x)/(1-x)^3. - _R. J. Mathar_, May 06 2008
%F a(n) = 16*n + a(n-1) - 2 with a(0)=0. - _Vincenzo Librandi_, Aug 05 2010
%F E.g.f.: (8*x^2 + 14*x)*exp(x). - _G. C. Greubel_, Jul 18 2017
%F From _Vaclav Kotesovec_, Aug 18 2018: (Start)
%F Sum_{n>=1} 1/a(n) = 2/9 + Pi/12 - log(2)/2.
%F Sum_{n>=1} (-1)^n/a(n) = 2/9 - Pi/(6*sqrt(2)) - log(2)/6 + log(1+sqrt(2))/(3*sqrt(2)). (End)
%t Table[2*n(4*n + 3), {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 26 2011 *)
%t LinearRecurrence[{3,-3,1},{0,14,44},80] (* _Harvey P. Dale_, Jun 05 2019 *)
%o (PARI) a(n)=2*n*(4*n+3) \\ _Charles R Greathouse IV_, Jun 17 2017
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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