A320660 Number of business cards required to build an origami level n Jerusalem cube.

12, 72, 672, 6048, 55488, 511872, 4738560, 43943424, 407890944, 3787941888, 35186122752, 326885842944, 3037038034944, 28217571901440, 262178452930560, 2436006721486848, 22634041833160704, 210303674768424960, 1954034324430913536, 18155901427591938048

Offset

0,1

Comments

The actual Jerusalem cube fractal cannot be built using a simple integer grid. However, one can create an approximate one by choosing the cube side length to be a Pell number (see link).

In practice, the first two terms represent the level 0 because they both consist of cubes (1 X 1 X 1 and 2 X 2 X 2, respectively). The "cross" shape appears at index 2, which is usually considered as the first iteration (for example, the "hole" shape in the Menger Sponge is visible at level 1).

The limit of a(n+1)/a(n) is equal to 2*(2+sqrt(7)) as n approaches infinity.

References

Eric Baird, L'art fractal, Tangente 150 (2013), 45.

Thomas Hull, Project Origami: Activities for Exploring Mathematics, A K Peters/CRC Press, 2006.

Links

Table of n, a(n) for n=0..19.

Eric Baird, The Jerusalem Cube

Malachi B-J. Brown, Business Card Origami

Robert Dickau, Cross Menger (Jerusalem) Cube Fractal

Origami Resource Center, Jerusalem Cube Fractal (Level 1)

Franck Ramaharo, An approximate Jerusalem square whose side equals a Pell number, arXiv:1801.00466 [math.CO], 2018.

Wikipedia, Cube de Jérusalem [In French]

Index entries for linear recurrences with constant coefficients, signature (12, -16, -80, -48)

Formula

a(n) = (3/14)*(7*(2 - 2*sqrt(2))^n + 7*(2 + 2*sqrt(2))^n + (21 - 5*sqrt(7))*(4 - 2*sqrt(7))^n + (21 + 5*sqrt(7))*(4 + 2*sqrt(7))^n).

a(n) = 12*a(n-1) - 16*a(n-2) - 80*a(n-3) - 48*a(n-4), n > 4.

G.f.: 12*(1 - 6*x + 8*x^3)/((1-4*x-4*x^2)*(1-8*x-12*x^2)) .

E.g.f.: (3/14)*(7*exp((2 - 2*sqrt(2))*x) + 7*exp((2 + 2*sqrt(2))*x) + (21 - 5*sqrt(7))*exp((4 - 2*sqrt(7))*x) + (21 + 5*sqrt(7))*exp((4 + 2*sqrt(7))*x)).

a(n) = 3*( A084128(n) -2*A239549(n) +3*A239549(n+1) ). - R. J. Mathar, Mar 06 2022

Example

a(2) = 672 because 456 business cards are needed for the squeleton and 216 more for the panels.

Mathematica

LinearRecurrence[{12, -16, -80, -48}, {12, 72, 672, 6048}, 20]

Prog

(Maxima) makelist((3/14)*(7*(2 - 2*sqrt(2))^n + 7*(2 + 2*sqrt(2))^n + (21 - 5*sqrt(7))*(4 - 2*sqrt(7))^n + (21 + 5*sqrt(7))*(4 + 2*sqrt(7))^n), n, 0, 20), ratsimp;

Crossrefs

At the n-th level, the cube side length is A000129(n+1), the squeleton requires 6*A239549(n+1) business cards, and each face requires A057087(n) units for the panels.

Cf. A212596 (Origami Menger sponge), A304960 (Origami Mosely snowflake sponge).

Sequence in context: A088166 A138402 A303505 * A367700 A108734 A143559

Adjacent sequences: A320657 A320658 A320659 * A320661 A320662 A320663

Keyword

nonn,easy

Author

Franck Maminirina Ramaharo, Oct 18 2018

Status

approved