A270488 Number of ordered ways to write n = x^2 + y*(y+1) + z*(z^2+1), where x, y and z are nonnegative integers.

1, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 3, 3, 3, 2, 2, 4, 2, 3, 1, 3, 3, 3, 3, 2, 2, 3, 2, 2, 2, 4, 6, 3, 3, 3, 1, 5, 3, 4, 4, 3, 4, 3, 2, 3, 3, 6, 2, 5, 2, 2, 5, 3, 3, 1, 4, 4, 4, 5, 3, 3, 5, 1, 1, 2, 3, 7, 4, 5, 4, 3, 3, 6, 2, 5, 4, 6, 2, 5, 4, 3

Offset: 0

Author

Zhi-Wei Sun, Mar 17 2016

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 5, 7, 19, 35, 54, 62, 63, 197, 285, 339, 479, 505, 917. Moreover, any integer n > 2 can be written as x^2 + y*(y+1) + z*(z^2+1), where x is a positive integer, and y and z are nonnegative integers.

We also guess that each n = 0,1,2,... can be expressed as x*(x+1)/2 + P(y,z) with x, y and z nonnegative integers, where P(y,z) is any of the polynomials y(y+1) + z^2*(z+1), y^2 + z*(z^2+2), y^2 + z*(z^2+7), y^2 + z*(z^2+z+2), y^2 + z*(z^2+2z+3), y^2 + z*(2z^2+z+1).

It is known that every n = 0,1,2,... can be written as x^2 + y*(y+1) + z*(z+1), where x, y and z are nonnegative integers.

Examples

			a(35) = 1 since 35 = 5^2 + 0*1 + 2*(2^2+1).
a(54) = 1 since 54 = 2^2 + 4*5 + 3*(3^2+1).
a(62) = 1 since 62 = 2^2 + 7*8 + 1*(1^2+1).
a(63) = 1 since 63 = 7^2 + 3*4 + 1*(1^2+1).
a(197) = 1 since 197 = 5^2 + 6*7 + 5*(5^2+1).
a(285) = 1 since 285 = 15^2 + 5*6 + 3*(3^2+1).
a(339) = 1 since 339 = 17^2 + 4*5 + 3*(3^2+1).
a(479) = 1 since 479 = 7^2 + 20*21 + 2*(2^2+1).
a(505) = 1 since 505 = 13^2 + 17*18 + 3*(3^2+1).
a(917) = 1 since 917 = 15^2 + 18*19 + 7*(7^2+1).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
• Zhi-Wei Sun, On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d), J. Number Theory 171(2017), 275-283.

Crossrefs

Cf. A000217, A000290, A000578, A002378, A002522, A262813, A262815, A262816, A270469.

Programs

Mathematica

SQ[x_]:=SQ[x]=IntegerQ[Sqrt[x]]
Do[r=0;Do[If[SQ[n-y(y+1)-z(z^2+1)],r=r+1],{y,0,(Sqrt[4n+1]-1)/2},{z,0,(n-y(y+1))^(1/3)}];Print[n," ",r];Continue,{n,0,80}]