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

0,1

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).

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).

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)

Example

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

Mathematica

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

Prog

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

Crossrefs

Sequence in context: A074578 A290719 A074576 * A362691 A138899 A147855

Adjacent sequences: A077241 A077242 A077243 * A077245 A077246 A077247

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 08 2002

Status

approved