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

0,2

Links

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

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

Example

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).

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 *)

Prog

(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*/

Crossrefs

Cf. A004018, A005875, A000118, A066535.

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

Sequence in context: A276178 A120369 A001665 * A265212 A295500 A168078

Adjacent sequences: A066533 A066534 A066535 * A066537 A066538 A066539

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 26 2009

Edited by R. J. Mathar, Oct 29 2009

Status

approved