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 A290342 Number of ways to write n as x^2 + 2*y^2 + z*(z+1)/2, where x is a nonnegative integer, and y and z are positive integers. 5
 0, 0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 2, 2, 2, 1, 2, 4, 3, 2, 4, 2, 4, 4, 3, 1, 4, 5, 2, 5, 1, 3, 6, 5, 2, 3, 6, 3, 9, 3, 1, 6, 3, 5, 4, 4, 6, 7, 3, 2, 5, 3, 6, 9, 6, 3, 7, 6, 2, 8, 5, 4, 8, 6, 3, 4, 6, 3, 12, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 COMMENTS Conjecture: a(n) > 0 for all n > 2. In other words, each n = 0,1,2,... can be written as x^2 + 2y*(y+2) + z*(z+3)/2 with x,y,z nonnegative integers. As pointed out by Sun in his 2007 paper in Acta Arith., a result of Jones and Pall implies that every n = 0,1,2,... can be written as x^2 + 2*(2y)^2 + z*(z+1)/2 with x,y,z nonnegative integers. Let a,c,e be positive integers, and let b,d,f be nonnegative integers with a-b, c-d, e-f all even. Suppose that a|b, c|d and e|f. The author studied in arXiv:1502.03056 when each nonnegative integer can be written as x*(a*x+b)/2 + y*(c*y+d)/2 + z*(e*z+f)/2 with x,y,z nonnegative integers, and conjectured that the answer is positive if (a,b,c,d,e,f) is among the following ten tuples (4,0,2,0,1,3), (4,0,2,0,1,5), (4,0,2,6,1,1), (4,0,2,6,2,0), (4,4,2,0,1,3), (4,8,2,0,1,1), (4,8,2,0,1,3), (4,12,2,0,1,1), (6,0,2,0,1,3), (6,6,2,0,1,3). See also A287616 and A286944 for related comments. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 B. W. Jones and G. Pall, Regular and semi-regular positive ternary quadratic forms, Acta Math. 70 (1939), 165-191. Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113. 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. EXAMPLE a(10) = 1 since 10 = 1^2 + 2*2^2 + 1*2/2. a(11) = 1 since 11 = 0^2 + 2*2^2 + 2*3/2. a(16) = 1 since 16 = 2^2 + 2*1^2 + 4*5/2. a(26) = 1 since 26 = 3^2 + 2*1^2 + 5*6/2. a(31) = 1 since 31 = 1^2 + 2*1^2 + 7*8/2. a(41) = 1 since 41 = 6^2 + 2*1^2 + 2*3/2. MATHEMATICA TQ[n_]:=TQ[n]=n>0&&IntegerQ[Sqrt[8n+1]] Do[r=0; Do[If[TQ[n-x^2-2y^2], r=r+1], {x, 0, Sqrt[n]}, {y, 1, Sqrt[(n-x^2)/2]}]; Print[n, " ", r], {n, 0, 70}] CROSSREFS Cf. A000217, A000290, A286944, A287616. Sequence in context: A023577 A188139 A134557 * A219842 A134264 A125181 Adjacent sequences:  A290339 A290340 A290341 * A290343 A290344 A290345 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jul 27 2017 STATUS approved

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Last modified June 19 13:00 EDT 2021. Contains 345129 sequences. (Running on oeis4.)