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%I #38 Feb 09 2017 03:02:50
%S 1,3,4,6,8,12,16,24,32,48,64,96,128,192,256,384,512,768,1024,1536,
%T 2048,3072,4096,6144,8192,12288,16384,24576,32768,49152,65536,98304,
%U 131072,196608,262144,393216,524288,786432,1048576,1572864,2097152
%N Spherical growth series for modular group.
%C 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
%D P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 156.
%H Reinhard Zumkeller, <a href="/A063759/b063759.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Gre#groups_modular">Index entries for sequences related to modular groups</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,2)
%F G.f.: (1+3*x+2*x^2)/(1-2*x^2).
%F a(n) = 2*a(n-2), n>2. - _Harvey P. Dale_, Oct 22 2011
%F a(2*n) = A151821(n+1); a(2*n+1) = A007283(n). - _Reinhard Zumkeller_, Dec 16 2013
%e For n = 2 the a(2) = 4 sequences are (1,2),(2,1),(2,3),(3,2). - _W. Edwin Clark_, Oct 17 2008
%e From _Joerg Arndt_, Nov 23 2012: (Start)
%e There are a(6) = 16 such words of length 6:
%e [ 1] [ 1 2 1 2 1 2 ]
%e [ 2] [ 1 2 1 2 3 2 ]
%e [ 3] [ 1 2 3 2 1 2 ]
%e [ 4] [ 1 2 3 2 3 2 ]
%e [ 5] [ 2 1 2 1 2 1 ]
%e [ 6] [ 2 1 2 1 2 3 ]
%e [ 7] [ 2 1 2 3 2 1 ]
%e [ 8] [ 2 1 2 3 2 3 ]
%e [ 9] [ 2 3 2 1 2 1 ]
%e [10] [ 2 3 2 1 2 3 ]
%e [11] [ 2 3 2 3 2 1 ]
%e [12] [ 2 3 2 3 2 3 ]
%e [13] [ 3 2 1 2 1 2 ]
%e [14] [ 3 2 1 2 3 2 ]
%e [15] [ 3 2 3 2 1 2 ]
%e [16] [ 3 2 3 2 3 2 ]
%e (End)
%t CoefficientList[Series[(1+3*x+2*x^2)/(1-2*x^2),{x,0,40}],x](* _Jean-François Alcover_, Mar 21 2011 *)
%t Join[{1},Transpose[NestList[{Last[#],2First[#]}&,{3,4},40]][[1]]] (* _Harvey P. Dale_, Oct 22 2011 *)
%o (Haskell)
%o import Data.List (transpose)
%o a063759 n = a063759_list !! n
%o a063759_list = concat $ transpose [a151821_list, a007283_list]
%o -- _Reinhard Zumkeller_, Dec 16 2013
%o (PARI) a(n)=([0,1; 2,0]^n*[1;3])[1,1] \\ _Charles R Greathouse IV_, Feb 09 2017
%Y Cf. A054886, A029744.
%Y The sequence (ternary strings) seems to be related to A029744 and A090989.
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_, Aug 14 2001
%E Information from A145751 included by _Joerg Arndt_, Dec 03 2012
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