|
%I #28 Jan 23 2020 01:48:41
%S 1,487,236681,115026479,55902632113,27168564180439,13203866289061241,
%T 6417051847919582687,3118673994222628124641,1515669144140349348992839,
%U 736612085378215560982395113,357991957824668622288095032079,173983354890703572216453203195281
%N Pell equation solutions (11*a(n))^2 - 122*b(n)^2 = -1 with b(n):=A097767(n), n >= 0.
%H Michael De Vlieger, <a href="/A097766/b097766.txt">Table of n, a(n) for n = 0..372</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 (486,-1).
%F G.f.: (1 + x)/(1 - 2*243*x + x^2).
%F a(n) = S(n, 2*243) + S(n-1, 2*243) = S(2*n, 2*sqrt(122)), with Chebyshev polynomials of the second 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, 11*i)/(11*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
%F a(n) = 486*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=487. - _Philippe Deléham_, Nov 18 2008
%F a(n) = (1/11)*sinh((2*n + 1)*arcsinh(11)). - _Bruno Berselli_, Apr 03 2018
%e (x,y) = (11*1=11;1), (5357=11*487;485), (2603491=11*236681;235709), ... give the positive integer solutions to x^2 - 122*y^2 =-1.
%t LinearRecurrence[{486, -1},{1, 487},12] (* _Ray Chandler_, Aug 12 2015 *)
%o (PARI) x='x+O('x^99); Vec((1+x)/(1-2*243*x+x^2)) \\ _Altug Alkan_, Apr 05 2018
%Y Cf. A097765 for S(n, 2*243).
%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
| 