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A063759 Spherical growth series for modular group. 9
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 (list; graph; refs; listen; history; text; internal format)
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
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
The sequence (ternary strings) seems to be related to A029744 and A090989.
Sequence in context: A298812 A299252 A299253 * A163978 A145751 A277099
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

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)