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A135918
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Genus of stage-n Menger sponge.
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3
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0, 5, 81, 1409, 26433, 514625, 10180161, 202704449, 4046898753, 80880453185, 1617148888641, 32339296372289, 646756476241473, 12934893915194945, 258695993426822721, 5173904789519844929, 103477975158264022593, 2069558538108217443905
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history;
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OFFSET
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0,2
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REFERENCES
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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.
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LINKS
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Colin Barker, Table of n, a(n) for n = 0..769
C. Mackeprang & K. Myers, Coloring Graphs on Sponges, Problem 11208, Amer. Math. Monthly 114 (November 2007), solutions p. 842.
Index entries for linear recurrences with constant coefficients, signature (29,-188,160).
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FORMULA
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a(n) = (21*20^n + 38*8^n - 59)/133.
From Garry John Tee, Feb 26 2015: (Start)
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).
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)
a(n) = 29*a(n-1)-188*a(n-2)+160*a(n-3). - Colin Barker, Feb 26 2015
G.f.: x*(64*x-5) / ((x-1)*(8*x-1)*(20*x-1)). - Colin Barker, Feb 26 2015
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EXAMPLE
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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.
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MAPLE
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A135918:=n->(21*20^n + 38*8^n - 59)/133: seq(A135918(n), n=0..20); # Wesley Ivan Hurt, Feb 27 2015
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MATHEMATICA
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Table[(21*20^n + 38*8^n - 59)/133, {n, 0, 16}] (* Michael De Vlieger, Feb 25 2015 *)
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PROG
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(PARI) concat(0, Vec(x*(64*x-5)/((x-1)*(8*x-1)*(20*x-1)) + O(x^100))) \\ Colin Barker, Feb 26 2015
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CROSSREFS
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Cf. A135919.
Sequence in context: A275347 A110257 A335177 * A115032 A278883 A307376
Adjacent sequences: A135915 A135916 A135917 * A135919 A135920 A135921
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KEYWORD
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easy,nonn
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AUTHOR
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Marc LeBrun, Dec 05 2007
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EXTENSIONS
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Reference to Tee (2015) by Garry John Tee, Feb 25 2015
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STATUS
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approved
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