A254661 Number of ways to write n as the sum of a triangular number, an even square and a second pentagonal number.

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

Offset: 0

Author

Zhi-Wei Sun, Feb 04 2015

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n. Also, a(n) = 1 only for n = 0, 1, 2, 4, 9, 11, 13, 20, 35, 69, 75, 188.

(ii) For each a = 2,3, any nonnegative integer n can be written as x(x+1)/2 + a*y^2 + z*(3*z+1)/2 with x,y,z nonnegative integers.

Compare part (i) of this conjecture with the conjecture in A160325.

Examples

			a(20) = 1 since 20 = 1*2/2 + 2^2 + 3*(3*3+1)/2.
a(35) = 1 since 35 = 7*8/2 + 0^2 + 2*(3*2+1)/2.
a(69) = 1 since 69 = 2*3/2 + 8^2 + 1*(3*1+1)/2.
a(75) = 1 since 75 = 9*10/2 + 2^2 + 4*(3*4+1)/2.
a(188) = 1 since 188 = 1*2/2 + 0^2 + 11*(3*11+1)/2.
		

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.

Crossrefs

Cf. A000217, A000290, A016742, A005449, A160325.

Programs

Mathematica

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
Do[r=0;Do[If[TQ[n-4y^2-z(3z+1)/2],r=r+1],{y,0,Sqrt[n/4]},{z,0,(Sqrt[24(n-4y^2)+1]-1)/6}];
Print[n," ",r];Label[aa];Continue,{n,0,100}]