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

0,3

Comments

Equals the row sums of triangle A232266.

Links

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

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.

Example

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; ...

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 *)

Prog

(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), ", "))

Crossrefs

Cf. A006456, A232266.

Sequence in context: A121337 A269069 A361036 * A342357 A279703 A206401

Adjacent sequences: A224363 A224364 A224365 * A224367 A224368 A224369

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2013

Status

approved