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A306561 Square numbers that are also central polygonal numbers (i.e., square numbers found in the Lazy Caterer's sequence).

%I A306561 #29 Jun 30 2025 19:35:26
%S A306561 1,4,16,121,529,4096,17956,139129,609961,4726276,20720704,160554241,
%T A306561 703893961,5454117904,23911673956,185279454481,812293020529,
%U A306561 6294047334436,27594051024016,213812329916329,937385441796001,7263325169820736,31843510970040004
%N A306561 Square numbers that are also central polygonal numbers (i.e., square numbers found in the Lazy Caterer's sequence).
%H A306561 G. C. Greubel, <a href="/A306561/b306561.txt">Table of n, a(n) for n = 1..1000</a>
%H A306561 J.S. Seneschal, <a href="/A306561/a306561.jpg">Relation To Square, Oblong, & Triangular Numbers On Multiplication Table</a>
%H A306561 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,34,-34,-1,1).
%F A306561 From _Alois P. Heinz_, Feb 23 2019: (Start)
%F A306561 G.f.: x*(1+3*x-22*x^2+3*x^3+x^4)/((1-x)*(1+6*x+x^2)*(1-6*x+x^2)).
%F A306561 a(n) = A006452(n)^2 for n >= 1.
%F A306561 { A000124 } intersect { A000290 }. (End)
%t A306561 LinearRecurrence[{1,34,-34,-1,1}, {1,4,16,121,529}, 25] (* _G. C. Greubel_, Apr 10 2019 *)
%o A306561 (PARI) my(x='x+O('x^25)); Vec(x*(1+3*x-22*x^2+3*x^3+x^4)/((1-x)*(1-34*x^2 +x^4))) \\ _G. C. Greubel_, Apr 10 2019
%o A306561 (PARI) lista(nn) = {for (n=0, nn, if (issquare(cpn = (n^2 + n) / 2 + 1), print1(cpn, ", ")););} \\ _Michel Marcus_, Apr 11 2019
%o A306561 (Magma) R<x>:=PowerSeriesRing(Integers(), 25); Coefficients(R!( x*(1+3*x-22*x^2+3*x^3+x^4)/((1-x)*(1-34*x^2 +x^4)) )); // _G. C. Greubel_, Apr 10 2019
%o A306561 (Sage) a=(x*(1+3*x-22*x^2+3*x^3+x^4)/((1-x)*(1-34*x^2 +x^4))).series(x, 25).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Apr 10 2019
%Y A306561 Cf. A000124, A000290, A006452.
%Y A306561 Cf. A000217, A002378.
%K A306561 nonn
%O A306561 1,2
%A A306561 _Moshe Monk_, Feb 23 2019
%E A306561 More terms from _Alois P. Heinz_, Feb 23 2019