A114052 x such that x^2 - 27*y^2 = 1.

1, 26, 1351, 70226, 3650401, 189750626, 9863382151, 512706121226, 26650854921601, 1385331749802026, 72010600134783751, 3743165875258953026, 194572614913330773601, 10114032809617941274226, 525735133485219615486151, 27328112908421802064005626, 1420536136104448487712806401

Offset

0,2

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 0..524

Tanya Khovanova, Recursive Sequences

John Robertson, Home page.

Index entries for linear recurrences with constant coefficients, signature (52,-1).

Formula

a(0) = 1, a(1) = 26 then a(n) = 52*a(n-1) - a(n-2). - Benoit Cloitre, Feb 03 2006

G.f.: (1 - 26x)/(1 - 52x + x^2). - Philippe Deléham, Nov 18 2008

a(n) = 1/2*(1+(26+15*sqrt(3))^(2*n))/(26+15*sqrt(3))^n. - Gerry Martens, May 30 2015

a(n) == 1 (mod 25). - Hugo Pfoertner, Feb 11 2024

Maple

f:= gfun:-rectoproc({a(n)=52*a(n-1)-a(n-2), a(0)=1, a(1)=26}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Jun 01 2015

Mathematica

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 *)
CoefficientList[Series[(1 - 26 x)/(1 - 52 x + x^2), {x, 0, 33}], x] (* Vincenzo Librandi, May 31 2015 *)
LinearRecurrence[{52, -1}, {1, 26}, 20] (* Harvey P. Dale, Jul 30 2017 *)

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=26; for(n=2, 30, a2=52*a1-a0; a0=a1; a1=a2; print1(a2, ", ")) \\ Benoit Cloitre, Feb 03 2006
(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

Crossrefs

Cf. A370188 (corresponding values of y, divided by 5).

Sequence in context: A160311 A220955 A106710 * A042303 A042300 A282884

Adjacent sequences: A114049 A114050 A114051 * A114053 A114054 A114055

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