A066536 Number of ways of writing n as a sum of n+1 squares.

1, 4, 12, 32, 90, 312, 1288, 5504, 22608, 88660, 339064, 1297056, 5043376, 19975256, 80027280, 321692928, 1291650786, 5177295432, 20748447108, 83279292960, 335056780464, 1351064867328, 5456890474248, 22063059606912

Offset: 0

Author

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

Keywords

• nonn

Examples

			There are a(2)=12 solutions (x,y,z) of 2=x^2+y^2+z^2: 3 permutations of (1,1,0), 3 permutations of (-1,-1,0) and 6 permutations of (1, -1,0).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A004018, A005875, A000118, A066535.

Cf. A122141, A166952. - Paul D. Hanna, Oct 26 2009

Programs

Mathematica

Join[{1}, Table[SquaresR[n+1, n], {n, 24}]]
(* Calculation of constants {d,c}: *) {1/r, Sqrt[s/(2*Pi*r^2*Derivative[0, 0, 2][EllipticTheta][3, 0, r*s])]} /. FindRoot[{s == EllipticTheta[3, 0, r*s], r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] == 1}, {r, 1/4}, {s, 5/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 16 2023 *)

PARI

{a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n));polcoeff(THETA3^(n+1), n)} /* Paul D. Hanna, Oct 26 2009*/

Formula

a(n) equals the coefficient of x^n in the (n+1)-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Oct 26 2009

a(n) is divisible by n+1: a(n)/(n+1) = A166952(n) for n>=0. - Paul D. Hanna, Oct 26 2009

a(n) ~ c * d^n / sqrt(n), where d = 4.13273137623493996302796465... (= 1/radius of convergence A166952), c = 0.70710538549959357505200... . - Vaclav Kotesovec, Sep 10 2014

Extensions

Edited by Dean Hickerson, Jan 12 2002

a(0) added by Paul D. Hanna, Oct 26 2009

Edited by R. J. Mathar, Oct 29 2009