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Number of ways to write n as an ordered sum of 6 squares of positive integers.
11

%I #22 Feb 04 2021 15:25:50

%S 1,0,0,6,0,0,15,0,6,20,0,30,15,0,60,12,15,60,31,60,30,60,90,36,86,60,

%T 120,120,15,180,141,60,165,140,180,186,120,180,285,156,126,360,255,

%U 216,270,260,390,240,262,420,426,360,210,540,530,216,540,540,480,600,300,600,825,312,576,840

%N Number of ways to write n as an ordered sum of 6 squares of positive integers.

%H Alois P. Heinz, <a href="/A340905/b340905.txt">Table of n, a(n) for n = 6..20000</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F G.f.: (theta_3(x) - 1)^6 / 64, where theta_3() is the Jacobi theta function.

%p b:= proc(n, t) option remember;

%p `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->

%p `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))

%p end:

%p a:= n-> b(n, 6):

%p seq(a(n), n=6..71); # _Alois P. Heinz_, Jan 31 2021

%t nmax = 71; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^6/64, {x, 0, nmax}], x] // Drop[#, 6] &

%Y Cf. A000141, A000290, A010052, A025430, A045848, A063691, A063725, A063730, A340481, A340906, A340915, A340946, A340947.

%Y Column k=6 of A337165.

%K nonn,look

%O 6,4

%A _Ilya Gutkovskiy_, Jan 31 2021