0,2

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

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

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).

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*/

Cf. A004018, A005875, A000118, A066535.

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

Sequence in context: A276178 A120369 A001665 * A393006 A265212 A295500

Adjacent sequences: A066533 A066534 A066535 * A066537 A066538 A066539

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

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

approved