A063759 Spherical growth series for modular group.

1, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152

Offset

0,2

Comments

Also number of sequences S of length n with entries in {1,..,q} where q = 3, satisfying the condition that adjacent terms differ in absolute value by exactly 1, see examples. - W. Edwin Clark, Oct 17 2008

References

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 156.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Index entries for sequences related to modular groups

Index entries for linear recurrences with constant coefficients, signature (0,2)

Formula

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

a(n) = 2*a(n-2), n>2. - Harvey P. Dale, Oct 22 2011

a(2*n) = A151821(n+1); a(2*n+1) = A007283(n). - Reinhard Zumkeller, Dec 16 2013

Example

For n = 2 the a(2) = 4 sequences are (1,2),(2,1),(2,3),(3,2). - W. Edwin Clark, Oct 17 2008
From Joerg Arndt, Nov 23 2012: (Start)
There are a(6) = 16 such words of length 6:
[ 1]   [ 1 2 1 2 1 2 ]
[ 2]   [ 1 2 1 2 3 2 ]
[ 3]   [ 1 2 3 2 1 2 ]
[ 4]   [ 1 2 3 2 3 2 ]
[ 5]   [ 2 1 2 1 2 1 ]
[ 6]   [ 2 1 2 1 2 3 ]
[ 7]   [ 2 1 2 3 2 1 ]
[ 8]   [ 2 1 2 3 2 3 ]
[ 9]   [ 2 3 2 1 2 1 ]
[10]   [ 2 3 2 1 2 3 ]
[11]   [ 2 3 2 3 2 1 ]
[12]   [ 2 3 2 3 2 3 ]
[13]   [ 3 2 1 2 1 2 ]
[14]   [ 3 2 1 2 3 2 ]
[15]   [ 3 2 3 2 1 2 ]
[16]   [ 3 2 3 2 3 2 ]
(End)

Mathematica

CoefficientList[Series[(1+3*x+2*x^2)/(1-2*x^2), {x, 0, 40}], x](* Jean-François Alcover, Mar 21 2011 *)
Join[{1}, Transpose[NestList[{Last[#], 2First[#]}&, {3, 4}, 40]][[1]]] (* Harvey P. Dale, Oct 22 2011 *)

Prog

(Haskell)
import Data.List (transpose)
a063759 n = a063759_list !! n
a063759_list = concat $ transpose [a151821_list, a007283_list]
-- Reinhard Zumkeller, Dec 16 2013
(PARI) a(n)=([0, 1; 2, 0]^n*[1; 3])[1, 1] \\ Charles R Greathouse IV, Feb 09 2017

Crossrefs

Cf. A054886, A029744.

The sequence (ternary strings) seems to be related to A029744 and A090989.

Sequence in context: A298812 A299252 A299253 * A163978 A145751 A277099

Adjacent sequences: A063756 A063757 A063758 * A063760 A063761 A063762

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Aug 14 2001

Extensions

Information from A145751 included by Joerg Arndt, Dec 03 2012

Status

approved