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A277792 Squares that are also pentagonal pyramidal numbers.

%I A277792 #35 Feb 16 2025 08:33:37
%S A277792 0,1,196,2601,15376,60025,181476,461041,1032256,2099601,3960100,
%T A277792 7027801,11861136,19193161,29964676,45360225,66846976,96216481,
%U A277792 135629316,187662601,255360400,342287001,452583076,591024721,763085376,975000625,1233835876,1547556921,1925103376,2376465001,2912760900
%N A277792 Squares that are also pentagonal pyramidal numbers.
%C A277792 Intersection of A000290 and A002411.
%H A277792 Daniel Mondot, <a href="/A277792/b277792.txt">Table of n, a(n) for n = 0..1000</a>
%H A277792 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalPyramidalNumber.html">Pentagonal Pyramidal Number</a>
%H A277792 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H A277792 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H A277792 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H A277792 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A277792 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A277792 O.g.f.: x*(1 + 189*x + 1250*x^2 + 1250*x^3 + 189*x^4 + x^5)/(1 - x)^7.
%F A277792 E.g.f.: x*(1 + 97*x + 336*x^2 + 256*x^3 + 60*x^4 + 4*x^5)*exp(x).
%F A277792 a(n) = a(-n).
%F A277792 a(n) = n^2*(2*n^2 - 1)^2.
%F A277792 a(n) = A000290(A007588(n)).
%F A277792 a(n) = A000290(n)*A000290(A056220(n)).
%F A277792 Sum_{n>=1} 1/a(n) = (2*Pi^2+9*sqrt(2)*Pi*cot(Pi/sqrt(2))+3*Pi^2*csc(Pi/sqrt(2))^2-24)/12 = 1.0055779712856...
%e A277792 a(2) = 196 because 196 = 14^2 is a perfect square and 196 = 7^2*(7 + 1)/2 is the 7th pentagonal pyramidal number.
%t A277792 Table[n^2 (2 n^2 - 1)^2, {n, 0, 30}]
%t A277792 LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,196,2601,15376,60025,181476},40] (* _Harvey P. Dale_, Nov 01 2024 *)
%o A277792 (Magma) [n^2*(2*n^2-1)^2: n in [0..30]]; // _Vincenzo Librandi_, Nov 01 2016
%Y A277792 Cf. A000290, A001653, A002411, A007588, A036353, A056220, A254333.
%K A277792 nonn,easy
%O A277792 0,3
%A A277792 _Ilya Gutkovskiy_, Oct 31 2016