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A082766
Series ratios converge alternately to sqrt(2) and 1+sqrt(1/2).
5
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
1,3
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
Haocong Song and Wen Wu, Hankel determinants of a Sturmian sequence, arXiv:2007.09940 [math.CO], 2020. See p. 4.
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
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 *)
PROG
(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
(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
CROSSREFS
Cf. A001333, A052542. See A119016 for another version.
Sequence in context: A373783 A281839 A136570 * A119016 A082958 A218495
KEYWORD
nonn
AUTHOR
Gary W. Adamson, May 24 2003
EXTENSIONS
Edited by Don Reble, Nov 04 2005
STATUS
approved