login
x such that x^2 - 23*y^2 = 1.
4

%I #51 Sep 21 2024 15:09:53

%S 1,24,1151,55224,2649601,127125624,6099380351,292643131224,

%T 14040770918401,673664360952024,32321848554778751,1550775066268428024,

%U 74404881332329766401,3569883528885560359224,171280004505174567476351,8217870332719493678505624

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

%C A Pellian equation (Pell's equation). - _Benoit Cloitre_, Feb 03 2006

%C Numbers n such that 23*(n^2-1) is a square. - _Vincenzo Librandi_, Nov 13 2010

%H Colin Barker, <a href="/A114051/b114051.txt">Table of n, a(n) for n = 0..594</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H John Robertson, <a href="https://web.archive.org/web/20190501092519/http://www.jpr2718.org/">Home page</a>.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (48,-1).

%F a(0)=1, a(1)=24 then a(n) = 48*a(n-1)-a(n-2). - _Benoit Cloitre_, Feb 03 2006

%F G.f.: (1-24*x)/(1-48*x+x^2). - _Philippe Deléham_, Nov 18 2008

%F 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

%F a(n) == 1 (mod 23). - _Hugo Pfoertner_, Feb 11 2024

%t LinearRecurrence[{48,-1},{1,24},20] (* _Harvey P. Dale_, Aug 19 2022 *)

%o (PARI) g(n,k) = for(y=0,n,x=k*y^2+1;if(issquare(x),print1(floor(sqrt(x))",")))

%o (PARI) a0=1;a1=24;for(n=2,30,a2=48*a1-a0;a0=a1;a1=a2;print1(a2,",")) \\ _Benoit Cloitre_, Feb 03 2006

%o (PARI) Vec((1-24*x)/(1-48*x+x^2) + O(x^20)) \\ _Colin Barker_, Jun 13 2015

%o (Magma) [n: n in [1..10000000] |IsSquare(23*(n^2-1))]; // _Vincenzo Librandi_, Nov 13 2010

%K easy,nonn

%O 0,2

%A _Cino Hilliard_, Feb 01 2006

%E More terms from _Benoit Cloitre_, Feb 03 2006

%E More terms from _Robert G. Wilson v_, Mar 17 2006