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 A097730 #44 Jan 17 2025 09:03:42
%S A097730 1,145,21169,3090529,451196065,65871534961,9616792908241,
%T A097730 1403985893068225,204972323595052609,29924555258984612689,
%U A097730 4368780095488158399985,637811969386012141785121,93116178750262284542227681,13594324285568907531023456305
%N A097730 Pell equation solutions (6*b(n))^2 - 37*a(n)^2 = -1 with b(n)=A097729(n), n >= 0.
%H A097730 Indranil Ghosh, <a href="/A097730/b097730.txt">Table of n, a(n) for n = 0..461</a>
%H A097730 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A097730 Giovanni Lucca, <a href="https://web.archive.org/web/20200716223631/http://forumgeom.fau.edu/FG2019volume19/FG201902.pdf">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.
%H A097730 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A097730 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (146,-1).
%F A097730 a(n) = S(n, 2*73) - S(n-1, 2*73) = T(2*n+1, sqrt(37))/sqrt(37), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
%F A097730 a(n) = ((-1)^n)*S(2*n, 12*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
%F A097730 G.f.: (1-x)/(1-146*x+x^2).
%F A097730 a(n) = 146*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=145. - _Philippe Deléham_, Nov 18 2008
%e A097730 (x,y) = (6,1), (882,145), (128766,21169), ... give the positive integer solutions to x^2 - 37*y^2 = -1.
%t A097730 LinearRecurrence[{146, -1},{1, 145},12] (* _Ray Chandler_, Aug 12 2015 *)
%o A097730 (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-146*x+x^2)) \\ _G. C. Greubel_, Aug 01 2019
%o A097730 (Magma) I:=[1,145]; [n le 2 select I[n] else 146*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Aug 01 2019
%o A097730 (Sage) ((1-x)/(1-146*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019
%o A097730 (GAP) a:=[1,145];; for n in [3..20] do a[n]:=146*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019
%Y A097730 Cf. A097729 for S(n, 146).
%Y A097730 Row 6 of array A188647.
%K A097730 nonn,easy
%O A097730 0,2
%A A097730 _Wolfdieter Lang_, Aug 31 2004