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%I #44 Nov 20 2022 12:11:15
%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 1664080,2692538,4356618,7049156,11405774,18454930,29860704,48315634,78176338,126491972
%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.
%C Also, number of sequences of length n with terms 1, 2, and 3, with no adjacent terms equal, and no three consecutive terms (1, 2, 3) or (3, 2, 1). - _Pontus von Brömssen_, Jan 03 2022
%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 = 0..999</a> [Offset changed to 0 by _Georg Fischer_, Mar 01 2022]
%H J. W. Cannon and 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/Con#coordseqs">Index entries for Coordination Sequences</a> [A layer sequence is a kind of coordination sequence. - _N. J. A. Sloane_, Nov 20 2022]
%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 (1,1).
%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+2) for n >= 2, with F(n) the n-th Fibonacci number (cf. A000045).
%F E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 1 - x. - _Stefano Spezia_, Apr 18 2022
%t Join[{1,3},2Fibonacci[Range[4,40]]] (* _Harvey P. Dale_, Jan 06 2012 *)
%o (PARI) my(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.
%Y Cf. A000045, A054888.
%K nonn,easy,nice
%O 0,2
%A Paolo Dominici (pl.dm(AT)libero.it), May 23 2000
%E Offset changed to 0 by _N. J. A. Sloane_, Jan 03 2022 at the suggestion of _Pontus von Brömssen_
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