A078989 Chebyshev sequence with Diophantine property.

1, 67, 4421, 291719, 19249033, 1270144459, 83810285261, 5530208682767, 364909962777361, 24078527334623059, 1588817894122344533, 104837902484740116119, 6917712746098725319321, 456464203340031130959067, 30119719707695955917979101, 1987445036504593059455661599

Offset

0,2

Comments

One fourth of bisection (even part) of A041024.

(4*a(n))^2 - 17*A078988(n)^2= -1 (Pell -1 equation, see A077232-3).

Links

Indranil Ghosh, Table of n, a(n) for n = 0..548

Tanya Khovanova, Recursive Sequences

Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.

Index entries for linear recurrences with constant coefficients, signature (66, -1).

Index entries for sequences related to Chebyshev polynomials.

Formula

G.f.: (1 + x)/(1 - 66*x + x^2).

a(n) = 66*a(n-1) - a(n-2) for n>=1, a(-1)=-1, a(0)=1.

a(n) = S(2*n, 2*sqrt(17)) = -i*((-1)^n)*T(2*n+1, 4*i)/4 = S(n, 66) + S(n-1, 66) with i^2=-1 and S(n, x), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.

a(n) = A041024(2*n)/4.

a(n) = (1/4)*sinh((2*n + 1)*arcsinh(4)). - Bruno Berselli, Apr 03 2018

Example

(x,y) = (4,1), (268,65), (17684,4289), ... give the positive integer solutions to x^2 - 17*y^2 =-1.

Mathematica

LinearRecurrence[{66, -1}, {1, 67}, 20] (* Bruno Berselli, Apr 03 2018 *)

Prog

(PARI) x='x+O('x^99); Vec((1+x)/(1-66*x+x^2)) \\ Altug Alkan, Apr 05 2018
(GAP) a:=[1, 67];; for n in [3..20] do a[n]:=66*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Apr 05 2018

Crossrefs

Cf. A097316 for S(n, 66).

Cf. A041024.

Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

Sequence in context: A120663 A261974 A328353 * A156121 A280004 A211961

Adjacent sequences: A078986 A078987 A078988 * A078990 A078991 A078992

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 10 2003

Status

approved