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A015223 Odd pentagonal pyramidal numbers.

%I A015223 #56 Mar 12 2025 13:21:27
%S A015223 1,75,405,1183,2601,4851,8125,12615,18513,26011,35301,46575,60025,
%T A015223 75843,94221,115351,139425,166635,197173,231231,269001,310675,356445,
%U A015223 406503,461041,520251,584325,653455,727833,807651,893101,984375
%N A015223 Odd pentagonal pyramidal numbers.
%C A015223 Also first bisection of A139757. - _Bruno Berselli_, Feb 13 2012
%H A015223 Bruno Berselli, <a href="/A015223/b015223.txt">Table of n, a(n) for n = 0..1000</a>
%H A015223 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalPyramidalNumber.html">Pentagonal Pyramidal Number</a>
%H A015223 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A015223 G.f.: (1 + 71*x + 111*x^2 + 9*x^3)/(1-x)^4. - _Colin Barker_, Feb 13 2012
%F A015223 a(n) = (2n+1)*(4n+1)^2 = A130656(4n+1). - _Bruno Berselli_, Feb 13 2012
%F A015223 From _Ant King_, Oct 23 2012: (Start)
%F A015223 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A015223 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 192.
%F A015223 Sum_{n>=0} 1/a(n) = (8*C - 2*Pi + Pi^2 - 4*log(2))/8, where C is Catalan's constant (A006752). (End)
%F A015223 E.g.f.: (1 + 74*x + 128*x^2 + 32*x^3)*exp(x). - _G. C. Greubel_, Nov 04 2017
%t A015223 Table[((n+1)^3+(n+1)^2)/2,{n,0,200,4}] (* _Vladimir Joseph Stephan Orlovsky_, May 21 2011 *)
%t A015223 CoefficientList[Series[(1 + 71 x + 111 x^2 + 9 x^3)/(1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 15 2013 *)
%o A015223 (PARI) a(n)=(2*n+1)*(4*n+1)^2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A015223 (Magma) [(2*n+1)*(4*n+1)^2: n in [0..50]]; // _G. C. Greubel_, Nov 04 2017
%Y A015223 Cf. A002411, A014799, A015224, A014800.
%K A015223 nonn,easy
%O A015223 0,2
%A A015223 _Mohammad K. Azarian_
%E A015223 More terms from _Erich Friedman_