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

0,2

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.

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

Example

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).

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}]]

Prog

(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 */

Crossrefs

Cf. A004018, A005875, A000118, A066536.

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

a(n^2) gives A361431.

Sequence in context: A065654 A002908 A004528 * A134455 A191700 A000643

Adjacent sequences: A066532 A066533 A066534 * A066536 A066537 A066538

Keyword

nonn

Author

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

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

Status

approved