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(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
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J. W. Cannon, 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.
<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 1).
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Number of n-th generation triangles in the tiling of the hyperbolic plane by triangles with angles {pi/2, pi/3, 0}.
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.
a(n) = a(n-1)+a(n-5) = a(n-2)+a(n-3), for n > 6.
a(1)=3 because exactly three triangles have generation 1, i.e. are adjacent to the triangle with generation 0.
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