A275648 Number of ordered ways to write n as x^2*(1+y^2+z^2)+w^2, where x is a positive integer and y,z,w are nonnegative integers with y <= z <= w.
1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 2, 4, 1, 1, 2, 3, 3, 3, 3, 2, 2, 1, 2, 4, 3, 3, 5, 3, 2, 2, 1, 4, 5, 2, 5, 4, 1, 2, 4, 4, 3, 3, 2, 5, 2, 1, 2, 6, 4, 4, 7, 4, 5, 3, 2, 4, 5, 2, 4, 5, 2, 4, 2, 6, 4, 4, 4, 4, 4, 1, 4, 7, 4, 4, 7, 1, 2, 3, 3
Offset: 1
Keywords
Examples
a(1) = 1 since 1 = 1^2*(1+0^2+0^2) + 0^2 with 0 = 0 = 0. a(2) = 1 since 2 = 1^2*(1+0^2+0^2) + 1^2 with 0 = 0 < 1. a(3) = 1 since 3 = 1^2*(1+0^2+1^2) + 1^2 with 0 < 1 = 1. a(6) = 1 since 6 = 1^2*(1+0^2+1^2) + 2^2 with 0 < 1 < 2. a(7) = 1 since 7 = 1^2*(1+1^2+1^2) + 2^2 with 1 = 1 < 2. a(11) = 1 since 11 = 1^2*(1+0^2+1^2) + 3^2 with 0 < 1 < 3. a(14) = 1 since 14 = 1^2*(1+0^2+2^2) + 3^2 with 0 < 2 < 3. a(15) = 1 since 15 = 1^2*(1+1^2+2^2) + 3^2 with 1 < 2 < 3. a(23) = 1 since 23 = 1^2*(1+2^2+3^2) + 3^2 with 2 < 3 = 3. a(38) = 1 since 38 = 1^2*(1+0^2+1^2) + 6^2 with 0 < 1 < 6. a(47) = 1 since 47 = 1^2*(1+1^2+3^2) + 6^2 with 1 < 3 < 6. a(71) = 1 since 71 = 1^2*(1+3^2+5^2) + 6^2 with 3 < 5 < 6. a(77) = 1 since 77 = 1^2*(1+2^2+6^2) + 6^2 with 2 < 6 = 6. a(143) = 1 since 143 = 1^2*(1+5^2+6^2) + 9^2 with 5 < 6 < 9. a(152) = 1 since 152 = 2^2*(1+0^2+1^2) + 12^2 with 0 < 1 < 12. a(191) = 1 since 191 = 1^2*(1+3^2+9^2) + 10^2 with 3 < 9 < 10. a(608) = 1 since 608 = 4^2*(1+0^2+1^2) + 24^2 with 0 < 1 < 24.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.
Programs
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Mathematica
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]] Do[r=0;Do[If[SQ[n-x^2*(1+y^2+z^2)],r=r+1],{x,1,Sqrt[n]},{y,0,Sqrt[(n-x^2)/(2x^2+1)]},{z,y,Sqrt[(n-x^2*(1+y^2))/(x^2+1)]}];Print[n," ",r];Continue,{n,1,80}]
Comments