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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, 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 06:59 EDT 2021. Contains 345452 sequences. (Running on oeis4.)