A283299 Number of ways to write 2*n + 1 as x^2 + 2*y^2 + 3*z^2 with x,y,z integers such that x + y + z is a square or twice a square.
1, 4, 3, 1, 6, 3, 1, 7, 1, 2, 8, 4, 4, 4, 3, 5, 4, 1, 4, 9, 3, 3, 9, 1, 4, 10, 3, 3, 11, 7, 4, 8, 5, 6, 7, 6, 2, 10, 3, 3, 14, 1, 2, 5, 3, 6, 12, 2, 4, 11, 3, 2, 5, 5, 7, 14, 6, 4, 6, 7, 4, 5, 4, 3, 13, 3, 3, 12, 3, 2, 15, 2, 2, 12, 3, 7, 4, 5, 4, 11, 8
Offset: 0
Keywords
Examples
a(0) = 1 since 2*0 + 1 = 1^2 + 2*0^2 + 3*0^2 with 1 + 0 + 0 = 1^2. a(3) = 1 since 2*3 + 1 = 2^2 + 2*0^2 + 3*(-1)^2 with 2 + 0 + (-1) = 1^2. a(8) = 1 since 2*8 + 1 = 3^2 + 2*(-2)^2 + 3*0^2 with 3 + (-2) + 0 = 1^2. a(17) = 1 since 2*17 + 1 = 0^2 + 2*(-2)^2 + 3^2 with 0 + (-2) + 3 = 1^2. a(41) = 1 since 2*41 + 1 = 9^2 + 2*(-1)^2 + 3*0^2 with 9 + (-1) + 0 = 2*2^2. a(128) = 1 since 2*128 + 1 = 3^2 + 2*10^2 + 3*(-4)^2 with 3 + 10 + (-4) = 3^2. a(197) = 1 since 2*197 + 1 = 12^2 + 2*(-2)^2 + 3*(-9)^2 with 12 + (-2) + (-9) = 1^2. a(372) = 1 since 2*372 + 1 = 22^2 + 2*3^2 + 3*(-9)^2 with 22 + 3 + (-9) = 4^2. a(764) = 1 since 2*764 + 1 = 18^2 + 2*(-23)^2 + 3*7^2 with 18 + (-23) + 7 = 2*1^2. a(3761) = 1 since 2*3761 + 1 = (-57)^2 + 2*31^2 + 3*28^2 with (-57) + 31 + 28 = 2*1^2. a(6408) = 1 since 2*6408 + 1 = (-22)^2 + 2*75^2 + 3*19^2 with (-22) + 75 + 19 = 2*6^2. a(6918) = 1 since 2*6918 + 1 = 100^2 + 2*9^2 + 3*35^2 with 100 + 9 + 35 = 12^2.
References
- L. E. Dickson, Modern Elementary Theory of Numbers, University of Chicago Press, Chicago, 1939. (See pages 112-113.)
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
- Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
- Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.
Programs
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Mathematica
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; TQ[n_]:=TQ[n]=SQ[n]||SQ[2n]; Do[r=0;Do[If[SQ[2n+1-3x^2-2y^2]&&TQ[(-1)^i*x+(-1)^j*y+(-1)^k*Sqrt[2n+1-3x^2-2y^2]],r=r+1],{x,0,Sqrt[(2n+1)/3]},{y,0,Sqrt[(2n+1-3x^2)/2]},{i,0,Min[x,1]},{j,0,Min[y,1]}, {k,0,Min[Sqrt[2n+1-3x^2-2y^2],1]}];Print[n," ",r];Continue,{n,0,80}]
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