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Number of integer solutions to a^2 + b^2 + c^2 + 8*d^2 = n.
9

%I #18 Sep 23 2019 09:05:49

%S 1,6,12,8,6,24,24,0,14,42,48,40,20,72,96,0,30,108,84,72,40,96,120,0,

%T 36,126,144,80,48,168,96,0,62,120,216,128,42,216,216,0,72,252,192,168,

%U 100,168,288,0,68,342,252,144,120,312,240,0,112,216,336,232,48,360,384

%N Number of integer solutions to a^2 + b^2 + c^2 + 8*d^2 = n.

%H Seiichi Manyama, <a href="/A236927/b236927.txt">Table of n, a(n) for n = 0..10000</a>

%H Liouville, J., <a href="http://sites.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1861_2_6_A41_0">Sur la forme X^2 + Y^2 + Z^2 + 8T^2</a>. Journal de mathématiques pures et appliquées 2e série, tome 6 (1861), p. 324-328.

%F G.f.: theta_3(q)^3*theta_3(q^8), where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Aug 01 2018

%p See A236924.

%Y Cf. A236924.

%Y For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1,2,3,4,5,6,7,8,12 see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Feb 15 2014