A082766 Series ratios converge alternately to sqrt(2) and 1+sqrt(1/2).

1, 1, 2, 3, 4, 7, 10, 17, 24, 41, 58, 99, 140, 239, 338, 577, 816, 1393, 1970, 3363, 4756, 8119, 11482, 19601, 27720, 47321, 66922, 114243, 161564, 275807, 390050, 665857, 941664, 1607521, 2273378, 3880899, 5488420, 9369319, 13250218, 22619537

Offset: 1

Author

Gary W. Adamson, May 24 2003

Keywords

• nonn

Comments

a(2n+2)/a(2n+1) converges to sqrt(2).

a(2n+1)/a(2n) converges to 1+sqrt(1/2).

a(n+2)/a(n) converges to 1+sqrt(2).

a(2n) is A001333, a(2n+1) is A052542.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Haocong Song and Wen Wu, Hankel determinants of a Sturmian sequence, arXiv:2007.09940 [math.CO], 2020. See p. 4.
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Crossrefs

Cf. A001333, A052542. See A119016 for another version.

Programs

Haskell

import Data.List (transpose)
a082766 n = a082766_list !! (n-1)
a082766_list = concat $ transpose [a052542_list, tail a001333_list]
-- Reinhard Zumkeller, Feb 24 2015

Mathematica

Rest[CoefficientList[Series[x (1 - x^2 + x) (x^2 + 1)/(1 - 2 x^2 - x^4), {x, 0, 50}], x]] (* G. C. Greubel, Nov 28 2017 *)
LinearRecurrence[{0,2,0,1},{1,1,2,3,4},50] (* Harvey P. Dale, Dec 15 2022 *)

PARI

x='x+O('x^30); Vec(x*(1+x-x^2)*(x^2+1)/(1-2*x^2-x^4)) \\ G. C. Greubel, Nov 28 2017

Formula

a(2n) = a(2n-1) + a(2n-2); a(2n+1) = a(2n) + a(2n-2)

O.g.f.: x*(1+x-x^2)*(x^2+1)/(1-2*x^2-x^4). - R. J. Mathar, Aug 08 2008

Extensions

Edited by Don Reble, Nov 04 2005