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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}.

%I A096231 #51 Jan 10 2025 04:39:18
%S A096231 1,3,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351,465,616,816,
%T A096231 1081,1432,1897,2513,3329,4410,5842,7739,10252,13581,17991,23833,
%U A096231 31572,41824,55405,73396,97229,128801,170625,226030,299426,396655,525456,696081
%N A096231 Number of n-th generation triangles in the tiling of the hyperbolic plane by triangles with angles {Pi/2, Pi/3, 0}.
%C A096231 Or, coordination sequence for (2,3,infinity) tiling of hyperbolic plane. - _N. J. A. Sloane_, Dec 29 2015
%C A096231 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.
%C A096231 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.
%C A096231 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
%H A096231 Robert Israel, <a href="/A096231/b096231.txt">Table of n, a(n) for n = 0..8110</a>
%H A096231 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.
%H A096231 Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, <a href="https://doi.org/10.9734/ajpas/2024/v26i12691">A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers</a>, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
%H A096231 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1).
%F A096231 a(n) = a(n-1) + a(n-5) = a(n-2) + a(n-3), for n > 6.
%F A096231 G.f.: (x+1)^2*(1+x+x^2) / (1-x^2-x^3).
%e A096231 a(1)=3 because exactly three triangles have generation 1, i.e., are adjacent to the triangle with generation 0.
%p A096231 f:= gfun:-rectoproc({a(n) = a(n-2)+a(n-3),
%p A096231 a(0)=1, a(1)=3, a(2)=5, a(3)=7, a(4)=9, a(5)=12}, a(n), remember):
%p A096231 seq(f(n),n=0..50); # _Robert Israel_, Jan 13 2016
%t A096231 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 *)
%t A096231 Join[{1, 3, 5}, LinearRecurrence[{0, 1, 1}, {7, 9, 12}, 50]] (* _Vincenzo Librandi_, Dec 30 2015 *)
%o A096231 (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
%o A096231 (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
%Y A096231 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.
%Y A096231 Equals A000931(n+10).
%K A096231 nonn,nice,easy
%O A096231 0,2
%A A096231 Bellovin, Kennedy, Stansifer, Wong (chrkenn(AT)bergen.org), Jul 29 2004
%E A096231 More terms from _Robert G. Wilson v_, Jul 31 2004