

A096231


Number of nth 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 n1 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) / (1X^2X^3) follows from the CannonWagreich 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



FORMULA

a(n) = a(n1) + a(n5) = a(n2) + a(n3), for n > 6.
G.f.: (x+1)^2*(1+x+x^2) / (1x^2x^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(n2)+a(n3),
a(0)=1, a(1)=3, a(2)=5, a(3)=7, a(4)=9, a(5)=12}, a(n), remember):


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(n1)+Self(n5): 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



KEYWORD

nonn,nice,easy


AUTHOR

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


EXTENSIONS



STATUS

approved



