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A352356 Number of ways to write 12*n + 5 as 2*x^2 + 5*y^2 + 9*z^2 + x*y*z, where x, y and z are nonnegative integers.

%I A352356 #14 Mar 14 2022 07:23:15
%S A352356 1,2,1,3,3,1,2,4,3,3,2,5,1,3,4,3,3,7,4,2,5,3,4,3,5,5,5,6,4,4,4,2,4,5,
%T A352356 6,3,6,5,6,5,4,5,6,7,4,4,6,4,7,6,5,3,3,8,3,7,7,4,5,7,5,6,6,8,4,1,4,7,
%U A352356 4,8,6,5,8,9,8,4,8,3,7,4,4,12,3,4,11,8,1,6,7,5,5,8,9,5,8,12,5,6,6,6,6
%N A352356 Number of ways to write 12*n + 5 as 2*x^2 + 5*y^2 + 9*z^2 + x*y*z, where x, y and z are nonnegative integers.
%C A352356 Conjecture: For each n = 0,1,2,... we can write 12*n + 5 as 2*x^2 + 5*y^2 + 9*z^2 + x*y*z with x,y,z nonnegative integers.
%C A352356 This has been verified for all n = 0..10^6.
%C A352356 It seems that a(n) = 1 only for n = 0, 2, 5, 12, 65, 86, 155, 338, 21030.
%H A352356 Zhi-Wei Sun, <a href="/A352356/b352356.txt">Table of n, a(n) for n = 0..10000</a>
%e A352356 a(0) = 1 with 12*0 + 5 = 5 = 2*0^2 + 5*1^2 + 9*0^2 + 0*1*0.
%e A352356 a(2) = 1 with 12*2 + 5 = 29 = 2*0^2 + 5*2^2 + 9*1^2 + 0*2*1.
%e A352356 a(5) = 1 with 12*5 + 5 = 65 = 2*3^2 + 5*1^2 + 9*2^2 + 3*1*2.
%e A352356 a(12) = 1 with 12*12 + 5 = 149 = 2*0^2 + 5*1^2 + 9*4^2 + 0*1*4.
%e A352356 a(65) = 1 with 12*65 + 5 = 785 = 2*1^2 + 5*9^2 + 9*6^2 + 1*9*6.
%e A352356 a(86) = 1 with 12*86 + 5 = 1037 = 2*6^2 + 5*1^2 + 9*10^2 + 6*1*10.
%e A352356 a(155) = 1 with 12*155 + 5 = 1865 = 2*2^2 + 5*6^2 + 9*13^2 + 2*6*13.
%e A352356 a(338) = 1 with 12*338 + 5 = 4061 = 2*20^2 + 5*6^2 + 9*13^2 + 20*6*13.
%e A352356 a(21030) = 1 with 12*21030 + 5 = 252365 = 2*32^2 + 5*126^2 + 9*39^2 + 32*126*39.
%t A352356 SQ[n_]:=IntegerQ[Sqrt[n]];
%t A352356 tab={};Do[r=0;Do[If[SQ[8(12n+5-5y^2-9z^2)+y^2*z^2]&&Mod[Sqrt[8(12n+5-5y^2-9z^2)+y^2*z^2]-y*z,4]==0,r=r+1],{y,0,Sqrt[(12n+5)/5]},{z,0,Sqrt[(12n+5-5y^2)/9]}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y A352356 Cf. A000290, A351723, A351617, A351902, A352259, A352286.
%K A352356 nonn
%O A352356 0,2
%A A352356 _Zhi-Wei Sun_, Mar 13 2022