A077244 Bisection (odd part) of Chebyshev sequence with Diophantine property.

3, 22, 173, 1362, 10723, 84422, 664653, 5232802, 41197763, 324349302, 2553596653, 20104423922, 158281794723, 1246149933862, 9810917676173, 77241191475522, 608118614128003, 4787707721548502, 37693543158260013, 296760637544531602, 2336391557197992803

Offset: 0

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• easy

Comments

3*a(n)^2 - 5*b(n)^2 = 7, with the companion sequence b(n)= A077243(n).

The even part is A077246(n) with Diophantine companion A077245(n).

Examples

			22 = a(1) = sqrt((5*A077243(1)^2 + 7)/3) = sqrt((5*17^2 + 7)/3) = sqrt(484) = 22.
		

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 (8,-1).

Programs

Magma

I:=[3,22]; [n le 2 select I[n] else 8*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 12 2015

Mathematica

 LinearRecurrence[{8, -1}, {3, 22}, 25] (* Vincenzo Librandi, Oct 12 2015 *)

PARI

Vec((3-2*x)/(1-8*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015

Formula

a(n)= (2*T(n+1, 4)+T(n, 4))/3, with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 4)= A001091(n).

G.f.: (3-2*x)/(1-8*x+x^2).

From Colin Barker, Oct 12 2015: (Start)

a(n) = (((4-sqrt(15))^n * (-10+3*sqrt(15)) + (4+sqrt(15))^n * (10+3*sqrt(15)))) / (2*sqrt(15)).

a(n) = 8*a(n-1) - a(n-2).

(End)