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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). 32
1, 3, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

Equivalently, coordination sequence for (3,3,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015

Equivalently, spherical growth series for modular group.

REFERENCES

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.

Index entries for sequences related to modular groups

FORMULA

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

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

MATHEMATICA

Join[{1, 3}, 2Fibonacci[Range[4, 40]]] (* Harvey P. Dale, Jan 06 2012 *)

PROG

(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

CROSSREFS

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.

Essentially the same as A006355.

Sequence in context: A114324 A265073 A265074 * A130578 A107068 A033541

Adjacent sequences:  A054883 A054884 A054885 * A054887 A054888 A054889

KEYWORD

nonn,easy,nice,changed

AUTHOR

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

STATUS

approved

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Last modified August 18 17:58 EDT 2017. Contains 290732 sequences.