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A306690
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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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1
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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, 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, 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
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OFFSET
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0,2
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COMMENTS
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Conjecture 1: a(n) > 0 for any nonnegative integer n.
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.
We have verified Conjectures 1 and 2 for n up to 2*10^6 and 10^6 respectively.
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LINKS
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EXAMPLE
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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.
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.
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.
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MATHEMATICA
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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];
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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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