A290472 Number of ways to write 6*n+1 as x^2 + 3*y^2 + 7*z^2, where x is a positive integer, and y and z are nonnegative integers.

1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 4, 1, 3, 1, 9, 1, 1, 2, 4, 3, 3, 3, 5, 1, 4, 2, 6, 3, 6, 1, 4, 2, 3, 1, 7, 3, 3, 3, 6, 2, 3, 2, 15, 2, 5, 2, 4, 2, 2, 7, 6, 3, 6, 2, 11, 3, 7, 3, 6, 4, 5, 2, 11, 4, 3, 1, 7, 3, 2, 4, 17, 2, 3, 3, 8, 2, 5, 7, 9, 4, 4, 2, 13, 1, 13, 1, 5, 4, 3, 4, 6, 7, 7, 3, 10, 4, 6, 3, 20, 3

Offset

0,4

Comments

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 2, 4, 5, 9, 10, 12, 14, 16, 17, 24, 30, 34, 66, 84, 86, 116, 124, 152, 286.

We also conjecture that {6n+5: n = 0,1,2,...} is a subset of {2x^2+3y^2+5z^2: x,y,z are nonnegative integers with y > 0}.

See A286885 for more similar conjectures.

In support of the first conjecture, a(n) > 1 for 286 < n <= 10^7. - Charles R Greathouse IV, Aug 04 2017

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

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.

Example

a(4) = 1 since 6*4+1 = 5^2 + 3*0^2 + 7*0^2.
a(5) = 1 since 6*5+1 = 2^2 + 3*3^2 + 7*0^2.
a(9) = 1 since 6*9+1 = 6^2 + 3*2^2 + 7*1^2.
a(116) = 1 since 6*116+1 = 9^2 + 3*14^2 + 7*2^2.
a(124) = 1 since 6*124+1 = 21^2 + 3*8^2 + 7*4^2.
a(152) = 1 since 6*152+1 = 19^2 + 3*10^2 + 7*6^2.
a(286) = 1 since 6*286+1 = 11^2 + 3*14^2 + 7*12^2.

Mathematica

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];
Do[r=0; Do[If[SQ[6n+1-3y^2-7z^2], r=r+1], {y, 0, Sqrt[(6n+1)/3]}, {z, 0, Sqrt[(6n+1-3y^2)/7]}]; Print[n, " ", r], {n, 0, 100}]

Prog

(PARI) a(n)=my(s=6*n+1, t); sum(z=0, sqrtint((s-1)\7), t=s-7*z^2; sum(y=0, sqrtint((t-1)\3), issquare(t-3*y^2))) \\ Charles R Greathouse IV, Aug 03 2017
(PARI) first(n)=my(v=vector(n+1), mx=6*n+1, s, t, u); for(x=1, sqrtint(mx), s=x^2; for(y=0, sqrtint((mx-s)\3), t=s+3*y^2; for(z=0, sqrtint((mx-t)\7), u=t+7*z^2; if(u%6==1, v[u\6+1]++)))); v \\ Charles R Greathouse IV, Aug 03 2017

Crossrefs

Cf. A000290, A286885, A286944, A287616, A290342.

Sequence in context: A047071 A124287 A253240 * A060240 A153734 A285554

Adjacent sequences: A290469 A290470 A290471 * A290473 A290474 A290475

Keyword

nonn

Author

Zhi-Wei Sun, Aug 03 2017

Status

approved