A381517 Perimeter of the Sierpiński carpet at iteration n.

4, 16, 80, 496, 3536, 26992, 212048, 1684720, 13442768, 107437168, 859182416, 6872514544, 54977282000, 439809752944, 3518452514384, 28147543587568, 225180119118032, 1801440264196720, 14411520047331152, 115292154179921392, 922337214843187664, 7378697662956950896, 59029581136289955920, 472236648588222693616

Offset: 0

Author

Jakub Buczak, Feb 26 2025

Keywords

• nonn
• easy

Comments

Carpet n has an overall size 3^n X 3^n and the perimeter here includes the perimeter of all holes within it.

Carpet n=0 is a unit square and has perimeter a(0) = 4.

Carpet n can be constructed by arranging 8 copies of carpet n-1 in a square with a hole in the middle,

X X X

X X

X X X

There are no gaps in each side so 2 sides of each n-1 are now not on the perimeter so a(n) = 8*a(n-1) - 16*3^(n-1).

An equivalent construction is to replace each of the 8^(n-1) unit squares of carpet n-1 with a 3 X 3 block of unit squares with a hole in the middle, so that a(n) = 3*a(n-1) + 4*8^(n-1).

A fractal is obtained by scaling the whole carpet down to a unit square and its scaled perimeter a(n)/3^n -> oo shows the perimeter is infinite even though the area is bounded.

Examples

			For n=0, a(0) = 4, the geometric representation is a square.
For n=3, a(3) = 496.
		

Links

• Jakub Buczak, Perimeter of the Sierpiński Carpet
• Michael Small, Brendan Florio, and Phillip Donald Fawell, The use of the perimeter area method to calculate the fractal dimension of aggregates, see section 3.2 equation (27) where a(n) = P_s(n+1) with scale factor g_1 = 1.
• Eric Weisstein's World of Mathematics, Sierpiński Carpet

Crossrefs

Cf. A001018, A271939, A293143, A365606.

Cf. A113210 (fractal dimension).

Programs

Python

a = lambda n: (4 * (4 * 3**n + 8**n)) // 5

Formula

a(n) = (4/5)*(4*3^n + 8^n).

a(n) = A365606(n+1) - 4.