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A286885 Number of ways to write 6*n+1 as x^2 + 3*y^2 + 54*z^2 with x,y,z nonnegative integers. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Conjecture: a(n) > 0 for all n = 0,1,2,....

In the a-file, we list the tuples (m,r,a,b,c) with 30 >= m > max{2,r} >= 0, 100 >= a >= b >= c > 0, gcd(a,b,c) = 1, and the form a*x^2+b*y^2+c*z^2 irregular, such that 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.

LINKS

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

Tomáš Hejda, 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, Zhi-Wei Sun, Arithmetic progressions represented by diagonal ternary quadratic forms, arXiv:1811.05855 [math.NT], 2018.

EXAMPLE

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.

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}]

CROSSREFS

Cf. A000290, A286944, A287616, A290342.

Sequence in context: A317990 A106030 A104888 * A254885 A108461 A321004

Adjacent sequences:  A286882 A286883 A286884 * A286886 A286887 A286888

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 02 2017

STATUS

approved

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Last modified June 25 12:01 EDT 2019. Contains 324352 sequences. (Running on oeis4.)