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%I #32 Mar 28 2022 08:16:12
%S 0,12,27,45,66,90,117,147,180,216,255,297,342,390,441,495,552,612,675,
%T 741,810,882,957,1035,1116,1200,1287,1377,1470,1566,1665,1767,1872,
%U 1980,2091,2205,2322,2442,2565,2691,2820,2952,3087,3225,3366,3510,3657,3807,3960
%N Generalized pentagonal numbers: a(n) = 12*n + 3*n*(n-1)/2.
%H G. C. Greubel, <a href="/A151542/b151542.txt">Table of n, a(n) for n = 0..1000</a>
%H Leo Tavares, <a href="/A151542/a151542.jpg">Illustration: Star Triangles</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = a(n-1) + 3*n + 9 (with a(0)=0). - _Vincenzo Librandi_, Nov 26 2010
%F G.f.: 3*x*(4 - 3*x)/(1 - x)^3. - _Arkadiusz Wesolowski_, Dec 24 2011
%F From _G. C. Greubel_, May 26 2017: (Start)
%F a(n) = 3*n*(n+7)/2.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F E.g.f.: (3/2)*(8*x + x^2)*exp(x). (End)
%F From _Amiram Eldar_, Feb 25 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 121/490.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/21 - 319/4410. (End)
%F a(n) = A003154(n+1) - A060544(n). - _Leo Tavares_, Mar 26 2022
%t s=0;lst={};Do[AppendTo[lst,s+=n],{n,12,6!,3}];lst (* _Vladimir Joseph Stephan Orlovsky_, Mar 05 2010 *)
%t LinearRecurrence[{3,-3,1}, {0,12,27}, 50] (* or *) With[{nn = 50}, CoefficientList[Series[(3/2)*(8*x + x^2)*Exp[x], {x, 0, nn}], x] Range[0, nn]!] (* _G. C. Greubel_, May 26 2017 *)
%o (PARI) x='x+O('x^50); concat([0], Vec(serlaplace((3/2)*(8*x + x^2)*exp(x)))) \\ _G. C. Greubel_, May 26 2017
%o (PARI) a(n)=(3*n^2+21*n)/2 \\ _Charles R Greathouse IV_, Jun 16 2017
%Y The generalized pentagonal numbers b*n + 3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.
%Y Cf. A003154, A060544.
%K easy,nonn
%O 0,2
%A _N. J. A. Sloane_, May 15 2009
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