A361197 - OEIS

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

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OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..96.

EXAMPLE

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.

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}];

CROSSREFS

Cf. A356770.

Sequence in context: A375094 A092938 A347744 * A320110 A068953 A362960

Adjacent sequences: A361194 A361195 A361196 * A361198 A361199 A361200

KEYWORD

nonn

AUTHOR

Luca Onnis, Mar 04 2023

STATUS

approved