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 A236924 Number of integer solutions to a^2 + 2*b^2 + 2*c^2 + 4*d^2 = n. 16
 1, 2, 4, 8, 8, 12, 16, 16, 24, 26, 24, 24, 32, 28, 32, 48, 24, 36, 52, 40, 48, 64, 48, 48, 96, 62, 56, 80, 64, 60, 96, 64, 24, 96, 72, 96, 104, 76, 80, 112, 144, 84, 128, 88, 96, 156, 96, 96, 96, 114, 124, 144, 112, 108, 160, 144, 192, 160, 120, 120, 192, 124, 128, 208, 24, 168, 192, 136, 144, 192, 192 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Olivia X. M. Yao, Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1--9. MR3141622. FORMULA Yao and Xia give an explicit formula for a(n) due to Liouville. G.f.: theta_3(q)*theta_3(q^2)^2*theta_3(q^4), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 03 2018 MAPLE with(LinearAlgebra); L:=proc(a, b, c, d, M) local n, hit, ul, vl, wl, xl, u, v, w, x, su, sv, sw, sx; hit:=Array(0..M); ul:=floor(sqrt(M/a)); vl:=floor(sqrt(M/b)); wl:=floor(sqrt(M/c)); xl:=floor(sqrt(M/d)); for u from -ul to ul do su:=a*u^2; for v from -vl to vl do sv:=su+b*v^2; for w from -wl to wl do sw:=sv+c*w^2; for x from -xl to xl do sx:=sw+d*x^2; if sx <= M then hit[sx]:=hit[sx]+1; fi; od; od; od; od; [seq(hit[n], n=0..M)]; end; L(1, 2, 2, 4, 60); CROSSREFS Cf. A000118, A320149, A320150, A320151, A320152, A320153. Sequence in context: A198346 A078750 A054785 * A266575 A260514 A123263 Adjacent sequences:  A236921 A236922 A236923 * A236925 A236926 A236927 KEYWORD nonn AUTHOR N. J. A. Sloane, Feb 14 2014 STATUS approved

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Last modified April 6 22:23 EDT 2020. Contains 333291 sequences. (Running on oeis4.)