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

1,2

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.

Links

Table of n, a(n) for n=1..100.

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

Example

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.

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]

Crossrefs

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

Sequence in context: A118975 A215644 A288777 * A087724 A152047 A289341

Adjacent sequences: A343500 A343501 A343502 * A343504 A343505 A343506

Keyword

nonn

Author

Zhi-Wei Sun, Apr 17 2021

Status

approved