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Number of representations of n as 2*w*x + x*y + y*z + z*u + u*x where w, x, y, z>0, u>=0 are integers.
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%I #10 Jul 28 2024 16:39:04

%S 0,0,0,1,1,4,3,11,4,21,10,30,15,51,14,71,28,86,36,119,36,150,55,178,

%T 59,219,68,269,91,278,105,367,110,396,116,469,153,504,162,587,190,608,

%U 210,752,199,759,253,906,261,879,296,1085,325,1068,322,1249,378,1239,406,1466,435,1425,431,1695,464

%N Number of representations of n as 2*w*x + x*y + y*z + z*u + u*x where w, x, y, z>0, u>=0 are integers.

%H Seiichi Manyama, <a href="/A375035/b375035.txt">Table of n, a(n) for n = 1..1000</a>

%H R. P. Agarwal, <a href="https://www.ias.ac.in/article/fulltext/pmsc/103/03/0269-0293">Lambert series and Ramanujan</a>, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293. see p. 275.

%F If n is an odd, a(n) = (sigma_2(n) + 4*sigma(n) - (4*n+1)*sigma_0(n))/8.

%o (PARI) a(n) = sum(w=1, n\2, sum(x=1, n\(2*w), sum(y=1, n\x, sum(z=1, n\y, sum(u=0, n\z, 2*w*x+x*y+y*z+z*u+u*x==n)))));

%Y Cf. A000005, A000203, A001157, A189835.

%K nonn

%O 1,6

%A _Seiichi Manyama_, Jul 28 2024