A077250 Bisection (odd part) of Chebyshev sequence with Diophantine property.

11, 103, 1019, 10087, 99851, 988423, 9784379, 96855367, 958769291, 9490837543, 93949606139, 930005223847, 9206102632331, 91131021099463, 902104108362299, 8929910062523527, 88396996516872971, 875040055106206183, 8662003554545188859, 85744995490345682407

Offset: 0

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• easy

Comments

a(n)^2 - 24*b(n)^2 = 25, with the companion sequence b(n) = A077249(n).

The even part is A077409(n) with Diophantine companion A077251(n).

Examples

			103 = a(1) = sqrt(24*A077249(1)^2 + 25) = sqrt(24*21^2 + 25) = sqrt(10609) = 103.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (10,-1).

Programs

Mathematica

CoefficientList[Series[(11 - 7 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)

PARI

a(n)= 2*polchebyshev(n+1,1,5)+polchebyshev(n,1,5) \\ Charles R Greathouse IV, Jun 11 2011

PARI

Vec((11-7*x)/(1-10*x+x^2) + O(x^30)) \\ Colin Barker, Jun 15 2015

Formula

a(n) = 10*a(n-1)- a(n-2), a(-1)=7, a(0)=11.

a(n) = 2*T(n+1, 5)+T(n, 5), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 5)= A001079(n).

a(n) = sqrt(25 + 24*A077249(n)^2).

G.f.: (11-7*x)/(1-10*x+x^2).