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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 15 15:16 EDT 2021. Contains 345049 sequences. (Running on oeis4.)