A114051 x such that x^2 - 23*y^2 = 1.

1, 24, 1151, 55224, 2649601, 127125624, 6099380351, 292643131224, 14040770918401, 673664360952024, 32321848554778751, 1550775066268428024, 74404881332329766401, 3569883528885560359224, 171280004505174567476351, 8217870332719493678505624

Offset

0,2

Comments

A Pellian equation (Pell's equation). - Benoit Cloitre, Feb 03 2006

Numbers n such that 23*(n^2-1) is a square. - Vincenzo Librandi, Nov 13 2010

Links

Colin Barker, Table of n, a(n) for n = 0..594

Tanya Khovanova, Recursive Sequences

John Robertson, Home page.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (48,-1).

Formula

a(0)=1, a(1)=24 then a(n) = 48*a(n-1)-a(n-2). - Benoit Cloitre, Feb 03 2006

G.f.: (1-24*x)/(1-48*x+x^2). - Philippe Deléham, Nov 18 2008

a(n) = T(n, 24) = (S(n, 48) - S(n-2, 48))/2, n >= 0, with Chebyshev's T- and S-polynomials (A049310 and A053120). - Wolfdieter Lang, Jul 02 2013

a(n) == 1 (mod 23). - Hugo Pfoertner, Feb 11 2024

Mathematica

LinearRecurrence[{48, -1}, {1, 24}, 20] (* Harvey P. Dale, Aug 19 2022 *)

Prog

(PARI) g(n, k) = for(y=0, n, x=k*y^2+1; if(issquare(x), print1(floor(sqrt(x))", ")))
(PARI) a0=1; a1=24; for(n=2, 30, a2=48*a1-a0; a0=a1; a1=a2; print1(a2, ", ")) \\ Benoit Cloitre, Feb 03 2006
(PARI) Vec((1-24*x)/(1-48*x+x^2) + O(x^20)) \\ Colin Barker, Jun 13 2015
(Magma) [n: n in [1..10000000] |IsSquare(23*(n^2-1))] - Vincenzo Librandi, Nov 13 2010

Crossrefs

Sequence in context: A130552 A160260 A268149 * A269092 A010562 A080775

Adjacent sequences: A114048 A114049 A114050 * A114052 A114053 A114054

Keyword

easy,nonn

Author

Cino Hilliard, Feb 01 2006

Extensions

More terms from Benoit Cloitre, Feb 03 2006

More terms from Robert G. Wilson v, Mar 17 2006

Status

approved