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A236923 Number of integer solutions to a^2 + b^2 + c^2 + 4*d^2 = n. 10
1, 6, 12, 8, 8, 36, 48, 16, 24, 78, 72, 24, 32, 84, 96, 48, 24, 108, 156, 40, 48, 192, 144, 48, 96, 186, 168, 80, 64, 180, 288, 64, 24, 288, 216, 96, 104, 228, 240, 112, 144, 252, 384, 88, 96, 468, 288, 96, 96, 342, 372, 144, 112, 324, 480, 144, 192, 480, 360, 120, 192, 372, 384, 208, 24, 504, 576, 136, 144, 576, 576, 144, 312, 444, 456, 248, 160, 576, 672 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..78.

Olivia X. M. Yao, Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1--9. MR3141622.

FORMULA

See Maple code.

G.f.: theta_3(q)^3*theta_3(q^4), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

MAPLE

with(numtheory);

s:=n-> if whattype(n) = integer then sigma(n) else 0; fi;

f:=proc(n) global s;

  if (n mod 4) = 0 then 8*s(n/4)-32*s(n/16)

elif (n mod 4) = 2 then 12*s(n/2)

elif (n mod 4) = 3 then 2*s(n)

else 6*s(n);

fi; end;

[seq(f(n), n=1..100)];

# a(0)=1 must be added separately

CROSSREFS

Cf. A097057, A236924.

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.

Sequence in context: A028659 A236930 A028643 * A028627 A162522 A292680

Adjacent sequences:  A236920 A236921 A236922 * A236924 A236925 A236926

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 14 2014

STATUS

approved

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Last modified September 23 12:15 EDT 2018. Contains 315274 sequences. (Running on oeis4.)