A096231 Number of n-th generation triangles in the tiling of the hyperbolic plane by triangles with angles {Pi/2, Pi/3, 0}.

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

Offset: 0

Author

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

Keywords

• nonn
• nice
• easy

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

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 0..8110
• J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257.
• Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
• Index entries for linear recurrences with constant coefficients, signature (0,1,1).

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).

Programs

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

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 *)

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

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).

Extensions

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