A097775 Pell equation solutions (14*a(n))^2 - 197*b(n)^2 = -1 with b(n) = A097776(n), n >= 0.

1, 787, 618581, 486203879, 382155630313, 300373839222139, 236093455472970941, 185569155627915937487, 145857120230086453893841, 114643510931692324844621539, 90109653735189937241418635813, 70826073192348358979430203127479, 55669203419532074967894898239562681

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Examples

			(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.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..345
• Tanya Khovanova, Recursive Sequences
• Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (786,-1).

Crossrefs

Cf. A097774 for S(n, 2*393).

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).

Programs

Mathematica

LinearRecurrence[{786, -1}, {1, 787}, 20] (* Harvey P. Dale, Dec 12 2017 *)

PARI

Vec((1+x)/(1-2*393*x+x^2) + O(x^100)) \\ Colin Barker, Apr 04 2015

Formula

G.f.: (1 + x)/(1 - 2*393*x + x^2).

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).

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.

a(n) = 786*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=787. - Philippe Deléham, Nov 18 2008

a(n) = (1/14)*sinh((2*n + 1)*arcsinh(14)). - Bruno Berselli, Apr 05 2018