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A236924 Number of integer solutions to a^2 + 2*b^2 + 2*c^2 + 4*d^2 = n. 16


%S 1,2,4,8,8,12,16,16,24,26,24,24,32,28,32,48,24,36,52,40,48,64,48,48,

%T 96,62,56,80,64,60,96,64,24,96,72,96,104,76,80,112,144,84,128,88,96,

%U 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

%N Number of integer solutions to a^2 + 2*b^2 + 2*c^2 + 4*d^2 = n.

%H Seiichi Manyama, <a href="/A236924/b236924.txt">Table of n, a(n) for n = 0..10000</a>

%H Olivia X. M. Yao, Ernest X. W. Xia, <a href="https://doi.org/10.1016/j.disc.2013.11.011">Combinatorial proofs of five formulas of Liouville</a>, Discrete Math. 318 (2014), 1--9. MR3141622.

%F Yao and Xia give an explicit formula for a(n) due to Liouville.

%F 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

%p with(LinearAlgebra);

%p L:=proc(a,b,c,d,M) local n,hit,ul,vl,wl,xl,u,v,w,x,su,sv,sw,sx;

%p hit:=Array(0..M); ul:=floor(sqrt(M/a)); vl:=floor(sqrt(M/b));

%p wl:=floor(sqrt(M/c)); xl:=floor(sqrt(M/d));

%p for u from -ul to ul do

%p su:=a*u^2;

%p for v from -vl to vl do

%p sv:=su+b*v^2;

%p for w from -wl to wl do

%p sw:=sv+c*w^2;

%p for x from -xl to xl do

%p sx:=sw+d*x^2;

%p if sx <= M then hit[sx]:=hit[sx]+1; fi;

%p od; od; od; od;

%p [seq(hit[n],n=0..M)];

%p end;

%p L(1,2,2,4,60);

%Y Cf. A000118, A320149, A320150, A320151, A320152, A320153.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Feb 14 2014

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Last modified August 7 17:44 EDT 2020. Contains 336278 sequences. (Running on oeis4.)