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A096231 Number of n-th generation triangles in the tiling of the hyperbolic plane by triangles with angles {pi/2, pi/3, 0}. 37
1, 3, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426, 396655, 525456, 696081 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Or, coordination sequence for (2,3,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015

The generation of a triangle is defined such that exactly one triangle has generation 0 and a triangle has generation n, n>0, if it is next to a triangle with generation n-1 but not to one with lower generation.

The recursions were found by examining empirical data and have not been proved to be accurate for all n. The generating function was found by assuming that the recursions were accurate; it can be calculated from either recursion. We created a specialized program in Java for finding the sequences of generations for triangles with angles {pi/p, pi/q, pi/r}, p, q, r > 1, that tile the Euclidean or hyperbolic plane; this program was used to calculate the sequence.

The g.f. (1+X)^2 * (1+X+X^2) / (1-X^2-X^3) follows from the Cannon-Wagreich paper, Prop. 3.1, so the g.f. and the recurrence are now a theorem, no longer conjectures, and the additional terms and the Mma program are now justified. - N. J. A. Sloane, Dec 29 2015

LINKS

Robert Israel, Table of n, a(n) for n = 0..8110

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

Index entries for linear recurrences with constant coefficients, signature (0, 1, 1).

FORMULA

a(n) = a(n-1)+a(n-5) = a(n-2)+a(n-3), for n > 6.

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

EXAMPLE

a(1)=3 because exactly three triangles have generation 1, i.e. are adjacent to the triangle with generation 0.

MAPLE

f:= gfun:-rectoproc({a(n) = a(n-2)+a(n-3),

a(0)=1, a(1)=3, a(2)=5, a(3)=7, a(4)=9, a(5)=12}, a(n), remember):

seq(f(n), n=0..50); # Robert Israel, Jan 13 2016

MATHEMATICA

CoefficientList[ Series[(x + 1)^2*(1 + x + x^2)/(1 - x^2 - x^3), {x, 0, 45}], x] (* Robert G. Wilson v, Jul 31 2004 *)

Join[{1, 3, 5}, LinearRecurrence[{0, 1, 1}, {7, 9, 12}, 50]] (* Vincenzo Librandi, Dec 30 2015 *)

PROG

(MAGMA) I:=[1, 3, 5, 7, 9, 12, 16]; [n le 7 select I[n] else Self(n-1)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Dec 30 2015

(PARI) a(n)=if(n>2, ([0, 1, 0; 0, 0, 1; 1, 1, 0]^n*[1; 3; 5])[1, 1], 1) \\ Charles R Greathouse IV, Feb 09 2017

CROSSREFS

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

Equals A000931(n+10).

Sequence in context: A194240 A265058 A265059 * A100432 A261033 A145341

Adjacent sequences:  A096228 A096229 A096230 * A096232 A096233 A096234

KEYWORD

nonn,nice,easy

AUTHOR

Bellovin, Kennedy, Stansifer, Wong (chrkenn(AT)bergen.org), Jul 29 2004

EXTENSIONS

More terms from Robert G. Wilson v, Jul 31 2004

STATUS

approved

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Last modified November 14 14:50 EST 2018. Contains 317208 sequences. (Running on oeis4.)