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A153111 Solutions of the Pell-like equation 1 + 6*A*A = 7*B*B, with A, B integers. 6
1, 25, 649, 16849, 437425, 11356201, 294823801, 7654062625, 198710804449, 5158826853049, 133930787374825, 3477041644892401, 90269151979827601, 2343520909830625225, 60841274503616428249, 1579529616184196509249, 41006928746285492812225 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

B is of the form B(i) = 26*B(i-1) - B(i-2) for B(0) = 1, B(1) = 25 (this sequence).

A is of the form A(i) = 26*A(i-1) - A(i-2) for A(0) = 1, A(1) = 27.

In general a Pell-like equation of the form 1 + X*A*A = (X + 1)*B*B has the solution A(i) = (4*X + 2)*A(i-1) - A(i-2), for A(0) = 1 and A(1) = (4*X + 3), and B(i) = (4*X + 2)*B(i-1) - B(i-2) for B(0) = 1 and B(1) = (4*X + 1).

Examples in OEIS:

  X = 1 gives A002315 for A(i) and A001653 for B(i);

  X = 2 gives A054320 for A(i) and A072256 for B(i);

  X = 3 gives A028230 for A(i) and A001570 for B(i);

  X = 4 gives A049629 for A(i) and A007805 for B(i);

  X = 5 gives A133283 for A(i) and A157014 for B(i);

  X = 6 gives A157461 for A(i) and this sequence for B(i).

Positive values of x (or y) satisfying x^2 - 26*x*y + y^2 + 24 = 0. - Colin Barker, Feb 20 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Luigi Cimmino, Algebraic relations for recursive sequences, arXiv:math/0510417 [math.NT], 2005-2008.

Jeroen Demeyer, Diophantine sets of polynomials over number fields, arXiv:0807.1970 [math.NT], 2008.

Franz Lemmermeyer, Conics - a Poor Man's Elliptic Curves, arXiv:math/0311306 [math.NT], 2003.

Index entries for linear recurrences with constant coefficients, signature (26,-1).

FORMULA

a(n) = 26*a(n-1) - a(n-2). - Colin Barker, Feb 20 2014

G.f.: -x*(x - 1) / (x^2 - 26*x + 1). - Colin Barker, Feb 20 2014

a(n) = (1/14)*(7 - sqrt(42))*(1 + (13 + 2*sqrt(42))^(2*n - 1))/(13 + 2*sqrt(42))^(n - 1). [Bruno Berselli, Feb 25 2014]

E.g.f.: (1/7)*(7*cosh(2*sqrt(42)*x) - sqrt(42)*sinh(2*sqrt(42)*x))*exp(13*x) - 1. - Franck Maminirina Ramaharo, Jan 07 2019

MATHEMATICA

CoefficientList[Series[(1 - x)/(x^2 - 26 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 22 2014 *)

LinearRecurrence[{26, -1}, {1, 25}, 20] (* Jean-François Alcover, Jan 07 2019 *)

PROG

(PARI) Vec(-x*(x-1)/(x^2-26*x+1) + O(x^100)) \\ Colin Barker, Feb 20 2014

(MAGMA) I:=[1, 25]; [n le 2 select I[n] else 26*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 22 2014

CROSSREFS

Cf. A002315, A001653, A054320, A072256, A001078, A028230, A001570, A049629, A007805, A133283, A140480.

Cf. similar sequences listed in A238379.

Sequence in context: A203341 A260048 A152256 * A097194 A180811 A318183

Adjacent sequences:  A153108 A153109 A153110 * A153112 A153113 A153114

KEYWORD

nonn,easy

AUTHOR

Ctibor O. Zizka, Dec 18 2008

EXTENSIONS

More terms from Philippe Deléham, Sep 19 2009; corrected by N. J. A. Sloane, Sep 20 2009

Additional term from Colin Barker, Feb 20 2014

STATUS

approved

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Last modified July 15 04:34 EDT 2020. Contains 335763 sequences. (Running on oeis4.)