A343503 Number of ways to write n as x*(3*x+1)/2 + y*(7*y+1)/2 + 2^k, where x and y are integers, and k is a nonnegative integer.

1, 2, 2, 3, 4, 6, 5, 5, 6, 4, 4, 5, 6, 4, 4, 8, 9, 6, 9, 8, 8, 6, 8, 7, 2, 7, 6, 6, 5, 7, 9, 8, 7, 10, 6, 11, 9, 9, 10, 6, 10, 9, 10, 6, 7, 10, 10, 6, 7, 6, 7, 7, 6, 7, 6, 11, 10, 9, 9, 9, 10, 10, 10, 9, 7, 7, 14, 8, 11, 9, 13, 11, 7, 13, 9, 7, 10, 8, 6, 7, 10, 11, 4, 9, 8, 12, 8, 11, 12, 6, 12, 11, 12, 13, 7, 12, 10, 11, 11, 9

Offset: 1

Author

Zhi-Wei Sun, Apr 17 2021

Keywords

• nonn

Comments

The author noted that a(n) > 0 for all n = 1..2*10^7. Giovanni Resta found that 8558169401 is the first value of n with a(n) = 0.

Examples

			a(1) = 1 with 1 = 0*(3*0+1)/2 + 0*(7*0+1)/2 + 2^0.
a(25) = 2, and 25 = 1*(3*1+1)/2 + 2*(7*2+1)/2 + 2^3 = (-2)*(3*(-2)+1)/2 + 1*(7*1+1)/2 + 2^4.
		

Links

• Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math. 63 (2020), 501-520.

Crossrefs

Cf. A000079, A001318, A270566, A343397, A343411, A343460.

Programs

Mathematica

PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]];
tab={};Do[r=0;Do[If[PenQ[n-2^k-x(7x+1)/2],r=r+1],{k,0,Log[2,n]},{x,-Floor[(Sqrt[56(n-2^k)+1]+1)/14],(Sqrt[56(n-2^k)+1]-1)/14}];tab=Append[tab,r],{n,1,100}];Print[tab]