A387549 Numbers k such that k^2 + 23 is twice a square.

3, 7, 25, 45, 147, 263, 857, 1533, 4995, 8935, 29113, 52077, 169683, 303527, 988985, 1769085, 5764227, 10310983, 33596377, 60096813, 195814035, 350269895, 1141287833, 2041522557, 6651912963, 11898865447, 38770189945, 69351670125, 225969226707, 404211155303

Offset

1,1

Comments

a(n) is the second number B of two numbers (A, B), such that the difference 2*A^2 - B^2 = 23. 2*A156066(n) gives the A value.

a(n) gives the x value solving the Diophantine equation x^2 + 23 = 2*y^2 or the Pell equation x^2 - 2*y^2 = -23. 2*A156066(n) gives the y value.

References

Leonard Eugene Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis, AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.

Links

Vladimir Pletser, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Formula

a(n) = a(n-1) + 4*A156066(n-1) + (-1)^(n-1)*(A048655(n-2) + A048655(n-3)) for n > 1 and with A048655(-1) = 3.

a(n) = 6*a(n-2) - a(n-4) with a(1) = 3, a(2) = 7, a(3) = 25, a(4) = 45.

a(n)*a(n+3) - a(n+1)*a(n+2) = -2*(34 + 14*(-1)^n).

a(n) = 2*A154138(n) + 1.

G.f.: x * (x+1) * (3*x^2+4*x+3) / ((x^2-2*x-1) * (x^2+2*x- 1)). - Elmo R. Oliveira, Apr 03 2026

Example

a(1) = 3 because 3^2 + 23 = 32 = 2*4^2.

Mathematica

LinearRecurrence[{0, 6, 0, -1}, {3, 7, 25, 45}, 30] (* Paolo Xausa, Sep 09 2025 *)

Crossrefs

Cf. A217975, A077446, A048655, A154138, A156066.

Sequence in context: A058781 A347614 A363535 * A100462 A161637 A329969

Adjacent sequences: A387546 A387547 A387548 * A387550 A387551 A387552

Keyword

nonn,easy

Author

Vladimir Pletser, Sep 01 2025

Status

approved