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