A361197 a(n) is the number of equations in the set {x^2 + 2y^2 = n, 2x^2 + 3y^2 = n, ..., k*x^2 + (k+1)*y^2 = n, ..., n*x^2 + (n+1)*y^2 = n} which admit at least one nonnegative integer solution.

1, 2, 3, 3, 3, 3, 3, 4, 4, 2, 5, 5, 3, 3, 3, 5, 4, 4, 5, 5, 5, 3, 3, 6, 4, 3, 6, 5, 5, 3, 5, 6, 4, 4, 4, 8, 3, 3, 5, 4, 6, 2, 5, 8, 6, 3, 3, 7, 6, 4, 6, 6, 4, 6, 3, 7, 4, 2, 7, 5, 6, 3, 6, 8, 3, 5, 5, 6, 7, 2, 5, 8, 4, 4, 6, 8, 4, 2, 6, 7, 8, 4, 5, 9, 3, 5, 4, 5, 6, 4, 6, 5, 4, 3, 4, 9

Offset: 1

Author

Luca Onnis, Mar 04 2023

Keywords

• nonn

Comments

Compared to the "linear" case given by A356770, the "quadratic" case given by this sequence has a more chaotic behavior.

a(n) >= 2 for all n > 1 since (n-1)*x^2 + n*y^2 = n and n*x^2 + (n+1)*y^2 = n always admit one integer solution (respectively (0,1) and (1,0)).

Conjecture: a(n) = 2 for infinitely many n.

Examples

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

Crossrefs

Cf. A356770.

Programs

Mathematica

b[m_] := m;
f[n_] := Table[Dimensions[Solve[b[k]*x^2 + b[k + 1]*y^2 == n, {x, y}, NonNegativeIntegers]][[1]], {k, 1, n}];