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A063759 Spherical growth series for modular group. 8
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

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-Fran├žois 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: A147606 A279083 A085147 * 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

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Last modified November 19 16:10 EST 2017. Contains 294936 sequences.