cp's OEIS Frontend

A066535 Number of ways of writing n as a sum of n squares.

%I A066535 #26 Mar 13 2023 08:09:53
%S A066535 1,2,4,8,24,112,544,2368,9328,34802,129064,491768,1938336,7801744,
%T A066535 31553344,127083328,509145568,2035437440,8148505828,32728127192,
%U A066535 131880275664,532597541344,2153312518240,8710505815360,35250721087168,142743029326162,578472382307304
%N A066535 Number of ways of writing n as a sum of n squares.
%H A066535 Alois P. Heinz, <a href="/A066535/b066535.txt">Table of n, a(n) for n = 0..1000</a>
%H A066535 John Holley-Reid and Jeremy Rouse, <a href="https://arxiv.org/abs/1910.01001">The number of representations of n as a growing number of squares</a>, arXiv:1910.01001 [math.NT], 2019.
%F A066535 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
%F A066535 a(n) ~ c * d^n / sqrt(n), where d = 4.13273137623493996302796465... (= 1/radius of convergence A166952), c = 0.2820942036723951157919967... . - _Vaclav Kotesovec_, Sep 12 2014
%e A066535 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).
%p A066535 b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
%p A066535       b(n, t-1) +2*add(b(n-j^2, t-1), j=1..isqrt(n))))
%p A066535     end:
%p A066535 a:= n-> b(n$2):
%p A066535 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 16 2014
%t A066535 Join[{1}, Table[SquaresR[n, n], {n, 24}]]
%o A066535 (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 */
%Y A066535 Cf. A004018, A005875, A000118, A066536.
%Y A066535 Cf. A122141, A166952. - _Paul D. Hanna_, Oct 25 2009
%Y A066535 a(n^2) gives A361431.
%K A066535 nonn
%O A066535 0,2
%A A066535 Peter Bertok (peter(AT)bertok.com), Jan 07 2002
%E A066535 Edited by _Dean Hickerson_, Jan 12 2002
%E A066535 a(0) added by _Paul D. Hanna_, Oct 25 2009
%E A066535 Edited by _R. J. Mathar_, Oct 29 2009