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Number of ways to write 6*n+1 as x^2 + 3*y^2 + 7*z^2, where x is a positive integer, and y and z are nonnegative integers.
4

%I #17 Jul 21 2023 06:53:33

%S 1,1,1,2,1,1,2,3,3,1,1,4,1,3,1,9,1,1,2,4,3,3,3,5,1,4,2,6,3,6,1,4,2,3,

%T 1,7,3,3,3,6,2,3,2,15,2,5,2,4,2,2,7,6,3,6,2,11,3,7,3,6,4,5,2,11,4,3,1,

%U 7,3,2,4,17,2,3,3,8,2,5,7,9,4,4,2,13,1,13,1,5,4,3,4,6,7,7,3,10,4,6,3,20,3

%N Number of ways to write 6*n+1 as x^2 + 3*y^2 + 7*z^2, where x is a positive integer, and y and z are nonnegative integers.

%C Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 2, 4, 5, 9, 10, 12, 14, 16, 17, 24, 30, 34, 66, 84, 86, 116, 124, 152, 286.

%C We also conjecture that {6n+5: n = 0,1,2,...} is a subset of {2x^2+3y^2+5z^2: x,y,z are nonnegative integers with y > 0}.

%C See A286885 for more similar conjectures.

%C In support of the first conjecture, a(n) > 1 for 286 < n <= 10^7. - _Charles R Greathouse IV_, Aug 04 2017

%H Zhi-Wei Sun, <a href="/A290472/b290472.txt">Table of n, a(n) for n = 0..10000</a>

%H Zhi-Wei Sun, <a href="https://doi.org/10.1007/s11425-015-4994-4">On universal sums of polygonal numbers</a>, Sci. China Math. 58 (2015), 1367-1396.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1502.03056">On universal sums x(ax+b)/2+y(cy+d)/2+z(ez+f)/2</a>, arXiv:1502.03056 [math.NT], 2015-2017.

%H Hai-Liang Wu and Zhi-Wei Sun, <a href="http://arxiv.org/abs/1707.06223">Some universal quadratic sums over the integers</a>, arXiv:1707.06223 [math.NT], 2017.

%e a(4) = 1 since 6*4+1 = 5^2 + 3*0^2 + 7*0^2.

%e a(5) = 1 since 6*5+1 = 2^2 + 3*3^2 + 7*0^2.

%e a(9) = 1 since 6*9+1 = 6^2 + 3*2^2 + 7*1^2.

%e a(116) = 1 since 6*116+1 = 9^2 + 3*14^2 + 7*2^2.

%e a(124) = 1 since 6*124+1 = 21^2 + 3*8^2 + 7*4^2.

%e a(152) = 1 since 6*152+1 = 19^2 + 3*10^2 + 7*6^2.

%e a(286) = 1 since 6*286+1 = 11^2 + 3*14^2 + 7*12^2.

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

%t Do[r=0;Do[If[SQ[6n+1-3y^2-7z^2],r=r+1],{y,0,Sqrt[(6n+1)/3]},{z,0,Sqrt[(6n+1-3y^2)/7]}];Print[n," ",r],{n,0,100}]

%o (PARI) a(n)=my(s=6*n+1,t); sum(z=0,sqrtint((s-1)\7), t=s-7*z^2; sum(y=0,sqrtint((t-1)\3), issquare(t-3*y^2))) \\ _Charles R Greathouse IV_, Aug 03 2017

%o (PARI) first(n)=my(v=vector(n+1),mx=6*n+1,s,t,u); for(x=1,sqrtint(mx), s=x^2; for(y=0,sqrtint((mx-s)\3), t=s+3*y^2; for(z=0,sqrtint((mx-t)\7), u=t+7*z^2; if(u%6==1, v[u\6+1]++)))); v \\ _Charles R Greathouse IV_, Aug 03 2017

%Y Cf. A000290, A286885, A286944, A287616, A290342.

%K nonn

%O 0,4

%A _Zhi-Wei Sun_, Aug 03 2017