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A054886 Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3,Pi/3,0) (this is the classical modular tessellation). 33

%I

%S 1,3,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,8362,

%T 13530,21892,35422,57314,92736,150050,242786,392836,635622,1028458,

%U 1664080,2692538,4356618,7049156,11405774,18454930,29860704,48315634

%N Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3,Pi/3,0) (this is the classical modular tessellation).

%C The layer sequence is the sequence of the cardinalities of the layers accumulating around a ( finite-sided ) polygon of the tessellation under successive side-reflections; see the illustration accompanying A054888.

%C Equivalently, coordination sequence for (3,3,infinity) tiling of hyperbolic plane. - _N. J. A. Sloane_, Dec 29 2015

%C Equivalently, spherical growth series for modular group.

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

%H G. C. Greubel, <a href="/A054886/b054886.txt">Table of n, a(n) for n = 1..1000</a>

%H J. W. Cannon, P. Wagreich, <a href="http://dx.doi.org/10.1007/BF01444714">Growth functions of surface groups</a>, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.

%H <a href="/index/Gre#groups_modular">Index entries for sequences related to modular groups</a>

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

%F a(n) = 2*F(n) for n>2, with F(n) the n-th Fibonacci number (cf. A000045 )

%t Join[{1,3},2Fibonacci[Range[4,40]]] (* _Harvey P. Dale_, Jan 06 2012 *)

%o (PARI) x='x+O('x^50); Vec((1+2*x+2*x^2+x^3)/(1-x-x^2)) \\ _G. C. Greubel_, Aug 06 2017

%Y Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

%Y Essentially the same as A006355.

%K nonn,easy,nice

%O 1,2

%A Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

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Last modified November 14 03:52 EST 2018. Contains 317159 sequences. (Running on oeis4.)