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

1, 4, 16, 121, 529, 4096, 17956, 139129, 609961, 4726276, 20720704, 160554241, 703893961, 5454117904, 23911673956, 185279454481, 812293020529, 6294047334436, 27594051024016, 213812329916329, 937385441796001, 7263325169820736, 31843510970040004

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,34,-34,-1,1).

Formula

From Alois P. Heinz, Feb 23 2019: (Start)

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)).

a(n) = A006452(n)^2 for n >= 1.

{ A000124 } intersect { A000290 }. (End)

Mathematica

LinearRecurrence[{1, 34, -34, -1, 1}, {1, 4, 16, 121, 529}, 25] (* G. C. Greubel, Apr 10 2019 *)

Prog

(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
(PARI) lista(nn) = {for (n=0, nn, if (issquare(cpn = (n^2 + n) / 2 + 1), print1(cpn, ", ")); ); } \\ Michel Marcus, Apr 11 2019
(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
(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

Crossrefs

Cf. A000124, A000290, A006452.

Sequence in context: A087335 A204573 A358880 * A226102 A094356 A322236

Adjacent sequences: A306558 A306559 A306560 * A306562 A306563 A306564

Keyword

nonn

Author

Moshe Monk, Feb 23 2019

Extensions

More terms from Alois P. Heinz, Feb 23 2019

Status

approved