A212596 Number of cards required to build a Menger sponge of level n in origami.
12, 192, 3456, 66048, 1296384, 25731072, 513048576, 10248388608, 204867108864, 4096536870912, 81924294967296, 1638434359738368, 32768274877906944, 655362199023255552, 13107217592186044416, 262144140737488355328, 5242881125899906842624
Offset: 0
Examples
12 cards (a(0)) are required for a single origami cube: 6 for the cube skeleton, and 6 for panels or possible links to other cubes.
Links
- CTRL Byte, 2010-04-13, Menger Sponge Construction [broken link]
- ELJJDX, Choux romanesco, vache qui rit et intégrales curvilignes, Am-stram-gram, ticket-ticket-bus-et-tram (French)
- Nick Hamblet, Σidiot's Blog, 2009-03-01, Counting Cards
- Michel Lucas, Défi 66 000 tickets (French)
- Jeannine Mosely, The Institute For Figuring, Business Card Menger Sponge
- Nicholas Rougeux, Mengermania, Instructions
- Wikipedia, Menger sponge
- Index entries for linear recurrences with constant coefficients, signature (28,-160).
Crossrefs
Cf. A082685.
Programs
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Maple
A212596:=n->4*(8^n + 2*20^n); seq(A212596(n), n=0..10); # Wesley Ivan Hurt, Apr 02 2014
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Mathematica
Table[4 (8^n + 2*20^n), {n, 10}] (* Wesley Ivan Hurt, Apr 02 2014 *)
Formula
a(n) = 4*(8^n + 2*20^n) = 2^(2*n+3)*5^n+2^(3*n+2).
a(n) = A082685(n)*3*4^(n+1).
From Colin Barker, Apr 10 2014: (Start)
a(n) = 28*a(n-1)-160*a(n-2).
G.f.: -12*(12*x-1) / ((8*x-1)*(20*x-1)). (End)
Extensions
More terms from Colin Barker, Apr 10 2014