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

Author

Cino Hilliard, Feb 01 2006

Keywords

• easy
• nonn

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).

Programs

Magma

[n: n in [1..10000000] |IsSquare(23*(n^2-1))]; // Vincenzo Librandi, Nov 13 2010

Mathematica

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

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

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

Extensions

More terms from Benoit Cloitre, Feb 03 2006

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