This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A097775 #32 Jan 23 2020 03:27:49
%S A097775 1,787,618581,486203879,382155630313,300373839222139,
%T A097775 236093455472970941,185569155627915937487,145857120230086453893841,
%U A097775 114643510931692324844621539,90109653735189937241418635813,70826073192348358979430203127479,55669203419532074967894898239562681
%N A097775 Pell equation solutions (14*a(n))^2 - 197*b(n)^2 = -1 with b(n) = A097776(n), n >= 0.
%H A097775 Colin Barker, <a href="/A097775/b097775.txt">Table of n, a(n) for n = 0..345</a>
%H A097775 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A097775 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 A097775 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A097775 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (786,-1).
%F A097775 G.f.: (1 + x)/(1 - 2*393*x + x^2).
%F A097775 a(n) = S(n, 2*393) + S(n-1, 2*393) = S(2*n, 2*sqrt(197)), 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 A097775 a(n) = ((-1)^n)*T(2*n+1, 14*i)/(14*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
%F A097775 a(n) = 786*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=787. - _Philippe Deléham_, Nov 18 2008
%F A097775 a(n) = (1/14)*sinh((2*n + 1)*arcsinh(14)). - _Bruno Berselli_, Apr 05 2018
%e A097775 (x,y) = (14*1=14;1), (11018=14*787;785), (8660134=14*618581;617009), ... give the positive integer solutions to x^2 - 197*y^2 =-1.
%t A097775 LinearRecurrence[{786, -1}, {1, 787}, 20] (* _Harvey P. Dale_, Dec 12 2017 *)
%o A097775 (PARI) Vec((1+x)/(1-2*393*x+x^2) + O(x^100)) \\ _Colin Barker_, Apr 04 2015
%Y A097775 Cf. A097774 for S(n, 2*393).
%Y A097775 Cf. similar sequences of the type (1/k)*sinh((2*n + 1)*arcsinh(k)): A002315 (k=1), A049629 (k=2), A097314 (k=3), A078989 (k=4), A097726 (k=5), A097729 (k=6), A097732 (k=7), A097735 (k=8), A097738 (k=9), A097741 (k=10), A097766 (k=11), A097769 (k=12), A097772 (k=13), this sequence (k=14).
%K A097775 nonn,easy
%O A097775 0,2
%A A097775 _Wolfdieter Lang_, Aug 31 2004