

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
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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 ab, cd, ef all even. Suppose that ab, cd and ef. 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

ZhiWei Sun, Table of n, a(n) for n = 0..10000
B. W. Jones and G. Pall, Regular and semiregular positive ternary quadratic forms, Acta Math. 70 (1939), 165191.
ZhiWei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103113.
ZhiWei Sun, On universal sums x(ax+b)/2+y(cy+d)/2+z(ez+f)/2, arXiv:1502.03056 [math.NT], 20152017.


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[nx^22y^2], r=r+1], {x, 0, Sqrt[n]}, {y, 1, Sqrt[(nx^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

ZhiWei Sun, Jul 27 2017


STATUS

approved



