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
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- J.S. Seneschal, Relation To Square, Oblong, & Triangular Numbers On Multiplication Table
- Index entries for linear recurrences with constant coefficients, signature (1,34,-34,-1,1).
Programs
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Magma
R
:=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 -
Mathematica
LinearRecurrence[{1,34,-34,-1,1}, {1,4,16,121,529}, 25] (* G. C. Greubel, Apr 10 2019 *) -
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 -
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
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.
Extensions
More terms from Alois P. Heinz, Feb 23 2019