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%I #29 Feb 28 2023 23:49:02
%S 0,6,18,40,126,196,288,550,726,936,1470,1800,2176,3078,3610,4200,5566,
%T 6348,7200,9126,10206,11368,13950,15376,16896,20230,22050,23976,28158,
%U 30420,32800,37926,40678,43560,49726,53016,56448,63750,67626,71656
%N Even pentagonal pyramidal numbers.
%H G. C. Greubel, <a href="/A015224/b015224.txt">Table of n, a(n) for n = 0..5000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalPyramidalNumber.html">Pentagonal Pyramidal Number</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 3, -3, 0, -3, 3, 0, 1, -1).
%F From _Ant King_, Oct 24 2012: (Start)
%F a(n) = a(n-1) +3*a(n-3) -3*a(n-4) -3*a(n-6) +3*a(n-7) +a(n-9) -a(n-10).
%F a(n) = 3*a(n-3) -3*a(n-6) +a(n-9) +192.
%F Sum_{n>=0} 1/a(n) = log(2)/2 + Pi/4 + 5*Pi^2/24 - 2 - C = 0.27217..., where C is Catalan's constant (A006752).
%F G.f.: 2*x*(3+6*x+11*x^2+34*x^3+17*x^4+13*x^5+11*x^6+x^7) / ((1-x)^4*(1+x +x^2)^3). (End)
%F a(n) = A002411(A004772(n+1)). - _Bruno Berselli_, Oct 24 2012
%t LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{0,6,18,40,126,196,288,550, 726,936},40] (* _Ant King_, Oct 19 2012 *)
%o (PARI) x='x+O('x^30); concat([0], Vec(2*x*(3+6*x+11*x^2+34*x^3 +17*x^4 +13*x^5+11*x^6+x^7)/((1-x)^4*(1+x +x^2)^3))) \\ _G. C. Greubel_, Aug 24 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x*(3+6*x+11*x^2+34*x^3+17*x^4+13*x^5+11*x^6+x^7)/((1-x)^4*(1+x +x^2)^3))); // _G. C. Greubel_, Aug 24 2018
%Y Cf. A002411, A014800, A015223, A014799, A006752.
%K nonn,easy
%O 0,2
%A _Mohammad K. Azarian_
%E More terms from _Patrick De Geest_, Jul 14 1999
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