A153111 - OEIS

%I #43 Mar 09 2024 10:29:02

%S 1,25,649,16849,437425,11356201,294823801,7654062625,198710804449,

%T 5158826853049,133930787374825,3477041644892401,90269151979827601,

%U 2343520909830625225,60841274503616428249,1579529616184196509249,41006928746285492812225

%N Solutions of the Pell-like equation 1 + 6*A*A = 7*B*B, with A, B integers.

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

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

%C 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).

%C Examples in the OEIS:

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

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

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

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

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

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

%C Positive values of x (or y) satisfying x^2 - 26*x*y + y^2 + 24 = 0. - _Colin Barker_, Feb 20 2014

%H Vincenzo Librandi, <a href="/A153111/b153111.txt">Table of n, a(n) for n = 1..200</a>

%H Luigi Cimmino, <a href="http://arxiv.org/abs/math/0510417">Algebraic relations for recursive sequences</a>, arXiv:math/0510417 [math.NT], 2005-2008.

%H Jeroen Demeyer, <a href="http://arxiv.org/abs/0807.1970">Diophantine sets of polynomials over number fields</a>, arXiv:0807.1970 [math.NT], 2008.

%H Franz Lemmermeyer, <a href="http://arxiv.org/abs/math/0311306">Conics - a Poor Man's Elliptic Curves</a>, arXiv:math/0311306 [math.NT], 2003.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (26,-1).

%F a(n) = 26*a(n-1) - a(n-2). - _Colin Barker_, Feb 20 2014

%F G.f.: -x*(x - 1) / (x^2 - 26*x + 1). - _Colin Barker_, Feb 20 2014

%F 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

%F 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

%t CoefficientList[Series[(1 - x)/(x^2 - 26 x + 1), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 22 2014 *)

%t LinearRecurrence[{26, -1}, {1, 25}, 20] (* _Jean-François Alcover_, Jan 07 2019 *)

%o (PARI) Vec(-x*(x-1)/(x^2-26*x+1) + O(x^100)) \\ _Colin Barker_, Feb 20 2014

%o (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

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

%Y Cf. similar sequences listed in A238379.

%K nonn,easy

%O 1,2

%A _Ctibor O. Zizka_, Dec 18 2008

%E More terms from _Philippe Deléham_, Sep 19 2009; corrected by _N. J. A. Sloane_, Sep 20 2009

%E Additional term from _Colin Barker_, Feb 20 2014