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

0,3

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.

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.

Example

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).

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}]

Crossrefs

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

Sequence in context: A083039 A106253 A078720 * A083898 A078314 A330894

Adjacent sequences: A270485 A270486 A270487 * A270489 A270490 A270491

Keyword

nonn

Author

Zhi-Wei Sun, Mar 17 2016

Status

approved