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

1, 785, 617009, 484968289, 381184458145, 299610499133681, 235493471134615121, 185097568701308351425, 145486453505757229604929, 114352167357956481161122769, 89880658056900288435412891505

Offset

0,2

Links

Michael De Vlieger, 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).

Formula

a(n) = S(n, 2*393) - S(n-1, 2*393) = T(2*n+1, sqrt(197))/sqrt(197), 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.

a(n) = ((-1)^n)*S(2*n, 28*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.

G.f.: (1-x)/(1-786*x+x^2).

a(n) = 786*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=785. - Philippe Deléham, Nov 18 2008

Example

(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.

Mathematica

CoefficientList[Series[(1-x)/(1-786*x+x^2), {x, 0, 20}], x] (* Michael De Vlieger, Apr 15 2019 *)
LinearRecurrence[{786, -1}, {1, 785}, 20] (* G. C. Greubel, Aug 01 2019 *)

Prog

(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-786*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(Magma) I:=[1, 785]; [n le 2 select I[n] else 786*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-786*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 785];; for n in [3..20] do a[n]:=786*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Crossrefs

Cf. A097774 for S(n, 786).

Sequence in context: A252389 A159896 A031734 * A031526 A108795 A097774

Adjacent sequences: A097773 A097774 A097775 * A097777 A097778 A097779

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved