A306383 Number of ways to write n as x*(2x+1) + y*(2y+1) + z*(2z+1), where x,y,z are nonnegative integers with x <= y <= z.

1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 2, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 0, 1, 1, 1, 0

Offset: 0

Author

Zhi-Wei Sun, Feb 11 2019

Keywords

• nonn

Comments

Conjecture 1: a(n) > 0 for any integer n > 138158.

We have verified this for n up to 2*10^6. Note that n*(2n+1) (n = 0,1,...) are the second hexagonal numbers (A014105).

Conjecture 2: Any integer n > 146858 can be written as the sum of three hexagonal numbers (A000384).

Conjecture 3: Any integer n > 33066 can be written as the sum of three pentagonal numbers (A000326).

Conjecture 4: Any integer n > 24036 can be written as the sum of three second pentagonal numbers (A005449).

Conjecture 5: Let N(1) = 114862, N(-1) = 166897, N(3) = 196987 and N(-3) = 273118. Then, for any r among 1, -1, 3 and -3, each integer n > N(r) can be written as x*(5x+r)/2 + y*(5y+r)/2 + z*(5z+r)/2 with x,y,z nonnegative integers.

We have verified Conjectures 2-5 for n up to 10^6.

Examples

			a(223595) = 1 with 223595 = 95*(2*95+1) + 200*(2*200+1) + 250*(2*250+1).
a(290660) = 1 with 290660 = 136*(2*136+1) + 149*(2*149+1) + 323*(2*323+1).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math., in press.

Crossrefs

Cf. A000217, A000326, A000384, A000566, A005449, A005475, A005476, A008443, A014105, A147875, A306382.

Programs

Mathematica

QQ[n_]:=QQ[n]=IntegerQ[Sqrt[8n+1]]&&Mod[Sqrt[8n+1],4]==1;
tab={};Do[r=0;Do[If[QQ[n-x(2x+1)-y(2y+1)],r=r+1],{x,0,(Sqrt[8n/3+1]-1)/4},{y,x,(Sqrt[4(n-x(2x+1))+1]-1)/4}];tab=Append[tab,r],{n,0,100}];Print[tab]