A255350 Number of ways to write n as a*(2a-1)+ b*(2b-1) + c*(2c+1) + d*(2d+1), where a,b,c,d are nonnegative integers with a <= b and c <= d.

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 2, 1, 3, 2, 1, 2, 2, 2, 4, 4, 1, 2, 3, 3, 3, 3, 2, 2, 4, 3, 3, 3, 2, 5, 4, 3, 3, 4, 3, 4, 5, 2, 3, 5, 3, 5, 5, 2, 5, 5, 3, 5, 4, 4, 5, 6, 5, 4, 4, 3, 4, 5, 5, 7, 7, 1, 5, 7, 4, 7, 7, 4, 3, 8, 5, 5, 6, 6, 5, 6, 4, 6, 6, 5, 10, 7, 3, 5, 8, 7, 9, 7, 4, 4, 9, 5, 4, 8

Offset: 0

Author

Zhi-Wei Sun, Feb 21 2015

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n. In other words, any nonnegative integer can be expressed as the sum of two hexagonal numbers and two second hexagonal numbers.

(ii) Each nonnegative integer can be written as the sum of two pentagonal numbers and two second pentagonal numbers.

We have verified parts (i) and (ii) of the conjecture for n up to 10^7 and 10^6 respectively.

Examples

			a(23) = 1 since 23 = 1*(2*1-1) + 1*(2*1-1) + 0*(2*0+1) + 3*(2*3+1).
a(68) = 1 since 68 = 1*(2*1-1) + 4*(2*4-1) + 1*(2*1+1) + 4*(2*4+1).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.
• Zhi-Wei Sun, On universal sums a*x^2+b*y^2+f(z), a*T_x+b*T_y+f(z) and a*T_x+b*y^2+f(z), arXiv:1502.03056 [math.NT], 2015.

Crossrefs

Cf. A000326, A000384, A005449, A014105.

Programs

Mathematica

HQ[n_]:=IntegerQ[Sqrt[8n+1]]&&Mod[Sqrt[8n+1],4]==1
Do[r=0;Do[If[HQ[n-x(2x-1)-y(2y-1)-z(2z+1)],r=r+1],{x,0,(Sqrt[4n+1]+1)/4},{y,x,(Sqrt[8(n-x(2x-1))+1]+1)/4},{z,0,(Sqrt[4(n-x(2x-1)-y(2y-1))+1]-1)/4}];
Print[n," ",r];Continue,{n,0,10000}]