A254885 Number of ways to write n as the sum of two squares and a positive triangular number.

1, 1, 2, 1, 2, 2, 2, 2, 1, 3, 4, 2, 1, 3, 3, 3, 2, 2, 5, 3, 3, 2, 5, 2, 2, 5, 2, 5, 3, 4, 4, 4, 3, 1, 6, 3, 5, 5, 3, 5, 5, 3, 2, 5, 3, 8, 5, 2, 3, 4, 5, 3, 8, 4, 7, 6, 3, 3, 4, 5, 5, 6, 3, 5, 7, 4, 4, 8, 2, 6, 9, 2, 6, 6, 6, 4, 4, 5, 6, 7, 5, 6, 6, 4, 4, 11, 4, 6, 5, 3, 9, 6, 5, 4, 11, 6, 3, 4, 3, 9

Offset: 1

Author

Zhi-Wei Sun, Feb 10 2015

Keywords

• nonn

Comments

We have shown that a(n) > 0 for all n > 0. In fact, if n is a positive triangular number T(x) = x*(x+1)/2, then n = 0^2 + 0^2 + T(x); if n > 0 is not a triangular number, then by Theorem 1(ii) of the reference of Sun in 2007, there are nonnegative integers a,b,c,u,v,w such that n = a^2 + b^2 + T(c) = u^2 + v^2 + T(w) with a + b odd and u + v even, hence c and w cannot both be zero.

This result is stronger than Euler's observation that any nonnegative integer can be written as the sum of two squares and a triangular number. We have also proved that any positive integer can be written as the sum of a positive square and two triangular numbers.

Examples

			a(4) = 1 since 4 = 0^2 + 1^2 + 2*3/2.
a(9) = 1 since 9 = 2^2 + 2^2 + 1*2/2.
a(13) = 1 since 13 = 1^2 + 3^2 + 2*3/2.
a(34) = 1 since 34 = 2^2 + 3^2 + 6*7/2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
• Zhi-Wei Sun, On universal sums a*x^2+b*y^2+c*f(z), a*T_x+b*T_y+f(z) and a*T_x+b*y^2+c*f(z), arXiv:1502.03056 [math.NT], 2015.

Crossrefs

Cf. A000217, A000290.

Programs

Mathematica

TQ[n_]:=n>0&&IntegerQ[Sqrt[8n+1]]
Do[r=0;Do[If[TQ[n-x^2-y^2],r=r+1],{x,0,Sqrt[n]},{y,0,x}];
Print[n," ",r];Continue,{n,1,10000}]