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%I #31 Apr 10 2024 11:03:21
%S 0,5,81,1409,26433,514625,10180161,202704449,4046898753,80880453185,
%T 1617148888641,32339296372289,646756476241473,12934893915194945,
%U 258695993426822721,5173904789519844929,103477975158264022593,2069558538108217443905
%N Genus of stage-n Menger sponge.
%D G. J. Tee, Tunnel numbers for fractal polyhedra, 1-3. Appendix A in the online version of H. Molina-Abral, P. Real, A. Nakamura & R. Klette, Connectivity calculus of fractal polyhedrons, Pattern Recognition 48 No. 4 (April 2015), 1146-1156.
%H Colin Barker, <a href="/A135918/b135918.txt">Table of n, a(n) for n = 0..769</a>
%H G. Korvin, <a href="https://doi.org/10.1007/978-3-031-46700-4_5">Menger Sponge Models</a>, Statistical Rock Physics, Earth and Environmental Sciences Library. Springer, Cham, 2024.
%H C. Mackeprang & K. Myers, <a href="http://www.jstor.org/stable/27642353">Coloring Graphs on Sponges, Problem 11208</a>, Amer. Math. Monthly 114 (November 2007), solutions p. 842.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (29,-188,160).
%F a(n) = (21*20^n + 38*8^n - 59)/133.
%F From _Garry John Tee_, Feb 26 2015: (Start)
%F For n>2, there is the 3-term recurrence relation a(n) - 28a(n-1) + 160a(n-2) = -59, and a(n) == 1 (mod 64).
%F Also, applying that recurrence relation (mod 10) shows that, in decimal arithmetic, the final digit of a(n) repeats in the cyclic order 5 1 9 3. (End)
%F a(n) = 29*a(n-1)-188*a(n-2)+160*a(n-3). - _Colin Barker_, Feb 26 2015
%F G.f.: x*(64*x-5) / ((x-1)*(8*x-1)*(20*x-1)). - _Colin Barker_, Feb 26 2015
%e a(0)=0 because a cube has genus 0. a(1)=5 because a cube with holes drilled through the faces meeting in the center has genus 5.
%p A135918:=n->(21*20^n + 38*8^n - 59)/133: seq(A135918(n), n=0..20); # _Wesley Ivan Hurt_, Feb 27 2015
%t Table[(21*20^n + 38*8^n - 59)/133, {n, 0, 16}] (* _Michael De Vlieger_, Feb 25 2015 *)
%o (PARI) concat(0, Vec(x*(64*x-5)/((x-1)*(8*x-1)*(20*x-1)) + O(x^100))) \\ _Colin Barker_, Feb 26 2015
%Y Cf. A135919.
%K easy,nonn
%O 0,2
%A _Marc LeBrun_, Dec 05 2007
%E Reference to Tee (2015) by _Garry John Tee_, Feb 25 2015
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