A078989 Chebyshev sequence with Diophantine property.

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

Offset: 0

Author

Wolfdieter Lang, Jan 10 2003

Keywords

• nonn
• easy

Comments

One fourth of bisection (even part) of A041024.

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

Examples

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

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.

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.

Programs

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

Mathematica

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

PARI

x='x+O('x^99); Vec((1+x)/(1-66*x+x^2)) \\ Altug Alkan, Apr 05 2018

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