A212596 -id:A212596

Search: a212596 -id:a212596

Displaying 1-2 of 2 results found.

page 1

A304960

Number of business cards required to build an origami level n Mosely snowflake sponge.

+10
1

12, 168, 2784, 48912, 874416, 15709488, 282620784, 5086424112, 91551884016, 1647915162288, 29662379171184, 533922356331312, 9610600070213616, 172990789545095088, 3113834153217961584, 56049014464954558512, 1008882258904338303216, 18159880652953870707888

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

REFERENCES

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

LINKS

Table of n, a(n) for n=0..17.

Franck Ramaharo, Level Two Mosely Snowflake Sponge

Origami Resource Center, Business Card Origami

Origami Resource Center, Mosely Snowflake Sponge Fractal (Level 3)

The Institute For Figuring, The Mosely Snowflake Sponge at the USC Libraries

The Institute For Figuring, USC Sponge Guide

Wikipedia, Mosely snowflake

Index entries for linear recurrences with constant coefficients, signature (23,-90)

FORMULA

a(n) = (108*18^n + 48*5^n)/13.

a(n) = 18*a(n-1) - 48*5^(n - 1);

a(n) = 23*a(n-1) - 90*a(n-2), with a(0) = 12 and a(1) = 168.

G.f.: - 12*(9*x - 1)/((5*x - 1)*(18*x - 1)).

E.g.f.: (108*exp(18*x) + 48*exp(5*x))/13.

EXAMPLE

a(1) = 168 because 108 business cards are needed for the squeleton and 60 more for the panels (see guide in links).

MAPLE

seq((108*18^n + 48*5^n)/13, n = 0 .. 50);

MATHEMATICA

LinearRecurrence[{23, -90}, {12, 168}, 50]

PROG

(Maxima) makelist((108*18^n + 48*5^n)/13, n, 0, 50);

CROSSREFS

Cf. A212596 (Origami Menger sponge).

KEYWORD

nonn,easy

AUTHOR

Franck Maminirina Ramaharo, May 22 2018

STATUS

approved

A320660

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

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).