A242823 Perimeter (rounded down) of Pi-shaped box fractal after n iterations.

1, 2, 5, 15, 39, 103, 269, 700, 1821, 4736, 12313, 32016, 83242, 216429, 562716, 1463063, 3803966, 9890311, 25714810, 66858508, 173832121, 451963515, 1175105140, 3055273364, 7943710747, 20653647942, 53699484649

Offset: 0

Author

Kival Ngaokrajang, May 23 2014

Keywords

• nonn

Comments

Let 13 boxes be placed into a 5 X 5 square grid, arranged in the shape of a capital letter Pi (see illustration). Also let the initial side length of a box = 1/28. The side length of a box after n iterations will be 1/(4*A005050(n)) i.e., 1/28, 1/140, 1/700, 1/3500, ... The sides count (any lengths) is 12*A001019(n), i.e., 12, 108, 972, 8748, ... The Hausdorff dimension = log(13)/log(5) = 1.593692641167... or A154265.

Links

• Kival Ngaokrajang, Illustration of initial terms
• Eric Weisstein's World of Mathematics, Box Fractal
• Wikipedia, Vicsek fractal

Crossrefs

Cf. A001019, A005050, A154265.

Programs

PARI

{a=28;b=1;print1(1,", "); for (n=2,50, b=b*0.2; a=(a*13-16*2^(n-1)-8); print1(floor(a*b/28),", "))}