A135918 Genus of stage-n Menger sponge.

0, 5, 81, 1409, 26433, 514625, 10180161, 202704449, 4046898753, 80880453185, 1617148888641, 32339296372289, 646756476241473, 12934893915194945, 258695993426822721, 5173904789519844929, 103477975158264022593, 2069558538108217443905

Offset: 0

Author

Marc LeBrun, Dec 05 2007

Keywords

• easy
• nonn

Examples

			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.
		

References

• 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.

Links

• Colin Barker, Table of n, a(n) for n = 0..769
• G. Korvin, Menger Sponge Models, Statistical Rock Physics, Earth and Environmental Sciences Library. Springer, Cham, 2024.
• 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).

Crossrefs

Cf. A135919.

Programs

Maple

A135918:=n->(21*20^n + 38*8^n - 59)/133: seq(A135918(n), n=0..20); # Wesley Ivan Hurt, Feb 27 2015

Mathematica

Table[(21*20^n + 38*8^n - 59)/133, {n, 0, 16}] (* Michael De Vlieger, Feb 25 2015 *)

PARI

concat(0, Vec(x*(64*x-5)/((x-1)*(8*x-1)*(20*x-1)) + O(x^100))) \\ Colin Barker, Feb 26 2015

Formula

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

Extensions

Reference to Tee (2015) by Garry John Tee, Feb 25 2015