A356770 a(n) is the number of equations in the set {x+2y=n, 2x+3y=n, ..., k*x+(k+1)*y=n, ..., n*x+(n+1)*y=n} which admit at least one nonnegative integer solution.

1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 6, 8, 7, 8, 8, 9, 8, 10, 8, 10, 10, 10, 9, 12, 10, 11, 11, 12, 10, 13, 11, 13, 12, 12, 12, 15, 12, 13, 13, 15, 12, 15, 13, 15, 15, 14, 13, 17, 14, 16, 15, 16, 14, 17, 15, 17, 16, 16, 15, 20, 15, 16, 17, 18, 17, 19, 16, 18, 17, 19, 16, 21, 17, 18, 19, 19

Offset: 1

Author

Luca Onnis, Aug 27 2022

Keywords

• nonn

Comments

a(n) = ceiling(2*(sqrt(n)-1)) + ceiling(A000005(n)/2).

Examples

			a(5) = 4. Consider the equations: x+2y=5, 2x+3y=5, 3x+4y=5, 4x+5y=5, 5x+6y=5. Only four of them admit at least one nonnegative integer solution, since 3x+4y=5 has no nonnegative integer solution.
		

Crossrefs

Cf. A000005.

Programs

Mathematica

b[m_] := m;
f[n_] := Table[Dimensions[Solve[b[k]*x + b[k + 1]*y == n, {x, y}, NonNegativeIntegers]][[1]], {k, 1, n}];
Flatten[Table[Dimensions[DeleteCases[f[k], 0]], {k, 1, 100}]]