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Number of ways to write n as w^2 + x^2 + y^2 + z^2 + 3*x*y*z, where w is a positive integer, and x,y,z are nonnegative integers with x <= y <= z.
6

%I #10 Mar 07 2022 14:51:24

%S 1,1,1,1,2,2,1,1,3,3,2,1,3,3,1,2,4,3,2,2,5,4,0,3,4,5,3,1,6,4,2,1,5,5,

%T 3,5,5,5,1,3,8,4,3,2,7,7,1,3,5,7,5,3,5,9,3,4,8,3,5,1,9,8,1,2,8,9,3,5,

%U 9,6,2,5,6,8,4,6,7,7,1,3,15,6,5,5,9,9,2,4,12,9,5,2,5,10,1,5,9,8,7,5

%N Number of ways to write n as w^2 + x^2 + y^2 + z^2 + 3*x*y*z, where w is a positive integer, and x,y,z are nonnegative integers with x <= y <= z.

%C Conjecture: We have a(n) > 0 except for n = 23. In other words, any positive integer n not equal to 23 can be written as w^2 + x^2 + y^2 + z^2 + 3*x*y*z, where w is a positive integer and x,y,z are nonnegative integers.

%C It seems that a(n) > 1 for all n > 695.

%H Zhi-Wei Sun, <a href="/A351902/b351902.txt">Table of n, a(n) for n = 1..10000</a>

%e a(60) = 1 with 60 = 2^2 + 1^2 + 1^2 + 6^2 + 3*1*1*6.

%e a(128) = 1 with 128 = 8^2 + 0^2 + 0^2 + 8^2 + 3*0*0*8.

%e a(303) = 1 with 303 = 11^2 + 1^2 + 1^2 + 12^2 + 3*1*1*12.

%e a(359) = 1 with 359 = 3^2 + 1^2 + 5^2 + 12^2 + 3*1*5*12.

%e a(383) = 1 with 383 = 11^2 + 1^2 + 3^2 + 12^2 + 3*1*3*12.

%e a(695) = 1 with 695 = 17^2 + 1^2 + 9^2 + 9^2 + 3*1*9*9.

%t SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];

%t tab={};Do[r=0;Do[If[SQ[n-x^2-y^2-z^2-3*x*y*z],r=r+1],{x,0,Sqrt[n/3]},{y,x,Sqrt[(n-x^2)/2]},{z,y,Sqrt[n-x^2-y^2]}];tab=Append[tab,r],{n,1,100}];Print[tab]

%Y Cf. A000118, A000290, A000378, A351723.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Feb 25 2022