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Number of ways of writing n as a sum of n+1 squares.
4

%I #19 Nov 16 2023 11:27:55

%S 1,4,12,32,90,312,1288,5504,22608,88660,339064,1297056,5043376,

%T 19975256,80027280,321692928,1291650786,5177295432,20748447108,

%U 83279292960,335056780464,1351064867328,5456890474248,22063059606912

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

%H Alois P. Heinz, <a href="/A066536/b066536.txt">Table of n, a(n) for n = 0..1000</a>

%F 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

%F a(n) is divisible by n+1: a(n)/(n+1) = A166952(n) for n>=0. - _Paul D. Hanna_, Oct 26 2009

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

%e 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).

%t Join[{1}, Table[SquaresR[n+1, n], {n, 24}]]

%t (* 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 *)

%o (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*/

%Y Cf. A004018, A005875, A000118, A066535.

%Y Cf. A122141, A166952. - _Paul D. Hanna_, Oct 26 2009

%K nonn

%O 0,2

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

%E Edited by _Dean Hickerson_, Jan 12, 2002

%E a(0) added by _Paul D. Hanna_, Oct 26 2009

%E Edited by _R. J. Mathar_, Oct 29 2009