A286885 Number of ways to write 6*n+1 as x^2 + 3*y^2 + 54*z^2 with x,y,z nonnegative integers.
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 3, 2, 3, 1, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 4, 4, 3, 2, 2, 4, 2, 3, 3, 3, 3, 3, 2, 2, 4, 3, 4, 1, 3, 2, 3, 4, 3, 3, 3, 3, 2, 3, 3, 2, 4, 3, 2, 3, 2
Offset: 0
Keywords
Examples
a(9) = 1 since 6*9 + 1 = 1^2 + 3*0^2 + 54*1^2. a(34) = 1 since 6*34 + 1 = 2^2 + 3*7^2 + 54*1^2. a(125) = 1 since 6*125 + 1 = 26^2 + 3*5^2 + 54*0^2. a(130) = 1 since 6*130 + 1 = 22^2 + 3*9^2 + 54*1^2. a(133) = 1 since 6*133 + 1 = 11^2 + 3*8^2 + 54*3^2. a(203) = 1 since 6*203 + 1 = 25^2 + 3*6^2 + 54*3^2.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
- Tomáš Hejda and Vítezslav Kala, Ternary quadratic forms representing arithmetic progressions, arXiv:1906.02538 [math.NT], 2019.
- Zhi-Wei Sun, Tuples (m,r,a,b,c) with 30 >= m > max{2,r} >= 0 and 100 >= a >= b >= c > 0, for which all the numbers m*n+r (n = 0,1,2,...) should be representable by a*x^2+b*y^2+c*z^2 with x,y,z integers.
- Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58 (2015), 1367-1396.
- Zhi-Wei Sun, On universal sums x(ax+b)/2+y(cy+d)/2+z(ez+f)/2, arXiv:1502.03056 [math.NT], 2015-2017.
- Hai-Liang Wu and Zhi-Wei Sun, Some universal quadratic sums over the integers, arXiv:1707.06223 [math.NT], 2017.
- Hai-Liang Wu and Zhi-Wei Sun, Arithmetic progressions represented by diagonal ternary quadratic forms, arXiv:1811.05855 [math.NT], 2018.
Programs
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Mathematica
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; table={};Do[r=0;Do[If[SQ[6n+1-3y^2-54z^2],r=r+1],{y,0,Sqrt[(6n+1)/3]},{z,0,Sqrt[(6n+1-3y^2)/54]}];table=Append[table,r],{n,0,70}]
Comments