%I #43 Feb 11 2024 11:18:54
%S 1,26,1351,70226,3650401,189750626,9863382151,512706121226,
%T 26650854921601,1385331749802026,72010600134783751,
%U 3743165875258953026,194572614913330773601,10114032809617941274226,525735133485219615486151,27328112908421802064005626,1420536136104448487712806401
%N x such that x^2 - 27*y^2 = 1.
%C A Pellian equation (Pell's equation). - _Benoit Cloitre_, Feb 03 2006
%H Robert Israel, <a href="/A114052/b114052.txt">Table of n, a(n) for n = 0..524</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/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (52,-1).
%F a(0) = 1, a(1) = 26 then a(n) = 52*a(n-1) - a(n-2). - _Benoit Cloitre_, Feb 03 2006
%F G.f.: (1 - 26x)/(1 - 52x + x^2). - _Philippe Deléham_, Nov 18 2008
%F a(n) = 1/2*(1+(26+15*sqrt(3))^(2*n))/(26+15*sqrt(3))^n. - _Gerry Martens_, May 30 2015
%F a(n) == 1 (mod 25). - _Hugo Pfoertner_, Feb 11 2024
%p f:= gfun:-rectoproc({a(n)=52*a(n-1)-a(n-2),a(0)=1,a(1)=26},a(n), remember):
%p map(f, [$0..40]); # _Robert Israel_, Jun 01 2015
%t A114052[n_] := 1/2(1 + (26 + 15 Sqrt[3])^(2 n))/(26 + 15 Sqrt[3])^n; Table[A114052[n] // FullSimplify, {n, 0, 20}] (* _Gerry Martens_, May 30 2015 *)
%t CoefficientList[Series[(1 - 26 x)/(1 - 52 x + x^2), {x, 0, 33}], x] (* _Vincenzo Librandi_, May 31 2015 *)
%t LinearRecurrence[{52,-1},{1,26},20] (* _Harvey P. Dale_, Jul 30 2017 *)
%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=26;for(n=2,30,a2=52*a1-a0;a0=a1;a1=a2;print1(a2,",")) \\ _Benoit Cloitre_, Feb 03 2006
%o (Magma) I:=[1,26]; [n le 2 select I[n] else 52*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, May 31 2015
%Y Cf. A370188 (corresponding values of y, divided by 5).
%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