A097726 - OEIS

%I #44 Jan 23 2020 03:51:07

%S 1,103,10505,1071407,109273009,11144775511,1136657829113,

%T 115927953794015,11823514629160417,1205882564220568519,

%U 122988198035868828521,12543590317094399940623,1279323224145592925115025,130478425272533383961791927,13307520054574259571177661529

%N Pell equation solutions (5*a(n))^2 - 26*b(n)^2 = -1 with b(n):=A097727(n), n >= 0.

%C a(-1) = -1. - _Artur Jasinski_, Feb 10 2010

%C 5*a(n) gives the x-values in the solution to the Pell equation x^2 - 26*y^2 = -1. - _Colin Barker_, Aug 24 2013

%H Vincenzo Librandi, <a href="/A097726/b097726.txt">Table of n, a(n) for n = 0..200</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (102,-1).

%F G.f.: (1 + x)/(1 - 102*x + x^2).

%F a(n) = S(n, 2*51) + S(n-1, 2*51) = S(2*n, 2*sqrt(26)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).

%F a(n) = ((-1)^n)*T(2*n+1, 5*i)/(5*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

%F a(n) = 102*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=103. - _Philippe Deléham_, Nov 18 2008

%F a(n) = (1/5)*sinh((2*n-1)*arcsinh(5)), n >= 1. - _Artur Jasinski_, Feb 10 2010

%e (x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 = -1.

%t Table[(1/5) Round[N[Sinh[(2 n - 1) ArcSinh[5]], 100]], {n, 1, 50}] (* _Artur Jasinski_, Feb 10 2010 *)

%t CoefficientList[Series[(1 + x)/(1 - 102 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Apr 13 2014 *)

%t LinearRecurrence[{102,-1},{1,103},20] (* _Harvey P. Dale_, Aug 20 2017 *)

%o (PARI) x='x+O('x^99); Vec((1+x)/(1-102*x+x^2)) \\ _Altug Alkan_, Apr 05 2018

%Y Cf. A097725 for S(n, 102).

%Y Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121. - _Artur Jasinski_, Feb 10 2010

%Y Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 31 2004

%E More terms from _Harvey P. Dale_, Aug 20 2017