A097729 - OEIS

%I #38 Aug 29 2024 21:11:38

%S 1,147,21461,3133159,457419753,66780150779,9749444593981,

%T 1423352130570447,207799661618691281,30337327244198356579,

%U 4429041977991341369253,646609791459491641554359,94400600511107788325567161,13781841064830277603891251147,2012054394864709422379797100301

%N Pell equation solutions (6*a(n))^2 - 37*b(n)^2 = -1 with b(n):=A097730(n), n >= 0.

%H Indranil Ghosh, <a href="/A097729/b097729.txt">Table of n, a(n) for n = 0..461</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 (146,-1).

%F G.f.: (1 + x)/(1 - 2*73*x + x^2).

%F a(n) = S(n, 2*73) + S(n-1, 2*73) = S(2*n, 2*sqrt(37)), 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, 6*i)/(6*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

%F a(n) = 146*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=147. - _Philippe Deléham_, Nov 18 2008

%F a(n) = (1/6)*sinh((2*n + 1)*arcsinh(6)). - _Bruno Berselli_, Apr 03 2018

%e (x,y) = (6,1), (882,145), (128766,21169), ... give the positive integer solutions to x^2 - 37*y^2 =-1.

%t LinearRecurrence[{146, -1}, {1, 147}, 20] (* _Harvey P. Dale_, Sep 24 2012 *)

%o (PARI) x='x+O('x^99); Vec((1+x)/(1-2*73*x+x^2)) \\ _Altug Alkan_, Apr 05 2018

%Y Cf. A097728 for S(n, 2*73).

%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 a(12)-a(13) from _Harvey P. Dale_, Sep 24 2012