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

0,2

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).

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A135919.

Sequence in context: A110257 A370197 A335177 * A115032 A278883 A307376

Adjacent sequences: A135915 A135916 A135917 * A135919 A135920 A135921

Keyword

easy,nonn,changed

Author

Marc LeBrun, Dec 05 2007

Extensions

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

Status

approved