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Number of nonnegative solutions to x^2 + y^2 + z^2 = n.
(Formerly M2265 N0895)
9

%I M2265 N0895 #23 Oct 14 2023 23:45:13

%S 1,3,3,1,3,6,3,0,3,6,6,3,1,6,6,0,3,9,6,3,6,6,3,0,3,9,12,4,0,12,6,0,3,

%T 6,9,6,6,6,9,0,6,15,6,3,3,12,6,0,1,9,15,6,6,12,12,0,6,6,6,9,0,12,12,0,

%U 3,18,12,3,9,12,6,0,6,9,18,7,3,12,6,0,6,15,9,9,6,12,15,0,3,21,18,6,0,6

%N Number of nonnegative solutions to x^2 + y^2 + z^2 = n.

%D A. Das and A. C. Melissinos, Quantum Mechanics: A Modern Introduction, Gordon and Breach, 1986, p. 48.

%D H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002102/b002102.txt">Table of n, a(n) for n = 0..10000</a>

%F Coefficient of q^k in (1/8)*(1 + theta_3(0, q))^3, or coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^3.

%t a[n_] := Module[{x, y, z, c}, For[x=c=0, x^2<=n, x++, For[y=0, x^2+y^2<=n, y++, If[IntegerQ[Sqrt[n-x^2-y^2]], c++ ]]]; c]

%t CoefficientList[Series[Sum[q^n^2, {n, 0, 12}], {q, 0, 150}]^3, q]

%o (PARI) Vec(sum(k=0,9,x^(k^2),O(x^100))^3) \\ _Charles R Greathouse IV_, Jun 13 2012

%Y First differences of A000606.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Dean Hickerson_, Oct 07 2001