A066535 Number of ways of writing n as a sum of n squares.

1, 2, 4, 8, 24, 112, 544, 2368, 9328, 34802, 129064, 491768, 1938336, 7801744, 31553344, 127083328, 509145568, 2035437440, 8148505828, 32728127192, 131880275664, 532597541344, 2153312518240, 8710505815360, 35250721087168, 142743029326162, 578472382307304

Offset: 0

Author

Peter Bertok (peter(AT)bertok.com), Jan 07 2002

Keywords

• nonn

Examples

			There are a(3) = 8 solutions (x,y,z) of 3 = x^2 + y^2 + z^2: (1,1,1), (-1,-1,-1), 3 permutations of (1,1,-1) and 3 permutations of (1,-1,-1).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• John Holley-Reid and Jeremy Rouse, The number of representations of n as a growing number of squares, arXiv:1910.01001 [math.NT], 2019.

Crossrefs

Cf. A004018, A005875, A000118, A066536.

Cf. A122141, A166952. - Paul D. Hanna, Oct 25 2009

a(n^2) gives A361431.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1) +2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 16 2014

Mathematica

Join[{1}, Table[SquaresR[n, n], {n, 24}]]

PARI

{a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n)); polcoeff(THETA3^n, n)} /* Paul D. Hanna, Oct 25 2009 */

Formula

a(n) equals the coefficient of x^n in the n-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Oct 25 2009

a(n) ~ c * d^n / sqrt(n), where d = 4.13273137623493996302796465... (= 1/radius of convergence A166952), c = 0.2820942036723951157919967... . - Vaclav Kotesovec, Sep 12 2014

Extensions

Edited by Dean Hickerson, Jan 12 2002

a(0) added by Paul D. Hanna, Oct 25 2009

Edited by R. J. Mathar, Oct 29 2009