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Number of ways to write n as u^4 + (v*(v+1)/2)^2 + (x*(3x+1)/2)^2 + (y*(5y+1)/2)^2 + (z*(9z+1)/2)^2, where u and v are nonnegative integers and x,y,z are integers.
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%I #16 Mar 11 2019 17:21:30

%S 1,3,3,1,2,5,4,1,1,4,6,4,1,3,5,2,2,6,6,3,5,8,6,2,2,9,14,9,2,9,14,7,2,

%T 5,10,12,9,6,8,7,5,9,10,6,4,10,10,4,1,4,12,11,5,4,10,6,5,5,5,8,8,7,8,

%U 5,1,7,11,5,3,5,8,5,3,1,6,10,4,4,6,4,1,8,8,8,6,7,11,6,1,2,10,8,3,2,7,6,1,4,8,9,4

%N Number of ways to write n as u^4 + (v*(v+1)/2)^2 + (x*(3x+1)/2)^2 + (y*(5y+1)/2)^2 + (z*(9z+1)/2)^2, where u and v are nonnegative integers and x,y,z are integers.

%C Conjecture 1: a(n) > 0 for any nonnegative integer n.

%C Conjecture 2: Each n = 0,1,2,... can be written as f(u,v,x,y,z) with u,v,x,y,z integers, where f is any of the following polynomials: u^4 + (v*(v+1)/2)^2 + (x*(3x+1)/2)^2 + (y*(5y+1)/2)^2 + (z*(5z+3)/2)^2, u^4 + (v*(v+1)/2)^2 + (x*(3x+1)/2)^2 + (y*(5y+1)/2)^2 + (z*(3z+2))^2, (u*(u+1)/2)^2 + (v*(3v+1)/2)^2 + (x*(5x+1)/2)^2 + (y*(5y+3)/2)^2 + (z*(3z+2))^2, (u*(u+1)/2)^2 + (v*(3v+1)/2)^2 + (x*(5x+1)/2)^2 + (y*(5y+3)/2)^2 + (z*(4z+3))^2, (u*(u+1)/2)^2 + (v*(3v+1)/2)^2 + (x*(5x+1)/2)^2 + (y*(5y+3)/2)^2 + (z*(9z+7)/2)^2.

%C We have verified Conjectures 1 and 2 for n up to 2*10^6 and 10^6 respectively.

%H Zhi-Wei Sun, <a href="/A306690/b306690.txt">Table of n, a(n) for n = 0..10000</a>

%e a(8) = 1 with 8 = 0^4 + (0*(0+1)/2)^2 + (1*(3*1+1)/2)^2 + ((-1)*(5*(-1)+1)/2)^2 + (0*(9*0+1)/2)^2.

%e a(2953) = 1 with 2953 = 6^4 + (8*(8+1)/2)^2 + (0*(3*0+1)/2)^2 + (0*(5*0+1)/2)^2 + (2*(9*2+1)/2)^2.

%e a(8953) = 1 with 8953 = 2^4 + (7*(7+1)/2)^2 + (6*(3*6+1)/2)^2 + ((-1)*(5*(-1)+1)/2)^2 + ((-4)*(9*(-4)+1)/2)^2.

%t t[x_]:=t[x]=(x(x+1)/2)^2; f[x_]:=f[x]=(x(5x+1)/2)^2; g[x_]:=g[x]=(x(9x+1)/2)^2; SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; PQ[n_]:=PQ[n]=SQ[n]&&SQ[24*Sqrt[n]+1];

%t tab={};Do[r=0;Do[If[PQ[n-k^4-t[x]-f[y]-g[z]],r=r+1],{k,0,n^(1/4)},{x,0,(Sqrt[8*Sqrt[n-k^4]+1]-1)/2},{y,-Floor[(Sqrt[40*Sqrt[n-k^4-t[x]]+1]+1)/10],(Sqrt[40*Sqrt[n-k^4-t[x]]+1]-1)/10},{z,-Floor[(Sqrt[72*Sqrt[n-k^4-t[x]-f[y]]+1]+1)/18],(Sqrt[72*Sqrt[n-k^4-t[x]-f[y]]+1]-1)/18}];tab=Append[tab,r],{n,0,100}];Print[tab]

%Y Cf. A000118, A000217, A000290, A000583, A001318, A057569, A306606, A306614.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Mar 05 2019