A224366 Number of compositions of n^2 into sums of squares.

1, 1, 2, 11, 124, 2870, 133462, 12477207, 2344649612, 885591183971, 672331353833716, 1025954712063362545, 3146790000180780110540, 19400015532276248131470280, 240398159948843792847457589388, 5987629866666297470033540284817068, 299759874416459708067727376075503706332

Offset: 0

Author

Paul D. Hanna, Apr 05 2013

Keywords

• nonn

Comments

Equals the row sums of triangle A232266.

Examples

			Illustrate a(n) = Sum_{k=1..n} A006456(n^2-k^2):
a(1) = 1 = 1;
a(2) = 2 = 1 + 1;
a(3) = 11 = 7 + 3 + 1;
a(4) = 124 = 88 + 30 + 5 + 1;
a(5) = 2870 = 2024 + 710 + 124 + 11 + 1;
a(6) = 133462 = 94137 + 33033 + 5767 + 502 + 22 + 1;
a(7) = 12477207 = 8800750 + 3088365 + 539192 + 46832 + 2024 + 43 + 1; ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..81

Crossrefs

Cf. A006456, A232266.

Programs

Maple

b:= proc(n) option remember; local i; if n=0 then 1
      else 0; for i while i^2<=n do %+b(n-i^2) od fi
    end:
a:= n-> b(n^2):
seq(a(n), n=0..17);  # Alois P. Heinz, Aug 12 2017

Mathematica

b[0] = 1; b[n_] := b[n] = Sum[b[n-k], {k, Select[Range[n], IntegerQ[ Sqrt[#]]&]}];
a[n_] := b[n^2];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jun 09 2018 *)

PARI

{a(n)=polcoeff(1/(1-sum(k=1,n,x^(k^2))+x*O(x^(n^2))),n^2)}
for(n=0,21,print1(a(n),", "))

PARI

{A006456(n)=polcoeff(1/(1-sum(k=1,sqrtint(n+1),x^(k^2))+x*O(x^n)),n)}
{a(n)=if(n==0,1,sum(k=1,n,A006456(n^2-k^2)))}
for(n=0,21,print1(a(n),", "))

Formula

a(n) = [x^(n^2)] 1/(1 - Sum_{k>=1} x^(k^2)).

a(n) = A006456(n^2).

a(n) = Sum_{k=1..n} A006456(n^2-k^2) for n>=1 with a(0)=1.