A365607 Number of degree 3 vertices in the n-Sierpinski carpet graph.

0, 40, 328, 2536, 19912, 158056, 1260616, 10073320, 80551624, 644308072, 5154149704, 41232252904, 329855188936, 2638833008488, 21110638558792, 168885031942888, 1351080025960648, 10808639518937704, 86469114085259080, 691752906483344872, 5534023233270575560, 44272185810376054120

Offset: 1

Author

Allan Bickle, Sep 12 2023

Keywords

• nonn
• easy

Comments

The level 0 Sierpinski carpet graph is a single vertex. The level n Sierpinski carpet graph is formed from 8 copies of level n-1 by joining boundary vertices between adjacent copies.

Examples

			The level 1 Sierpinski carpet graph is an 8-cycle, which has 8 degree 2 vertices and 0 degree 3 or 4 vertices.  Thus a(1) = 0.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..1000
• Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
• Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
• Eric Weisstein's World of Mathematics, Sierpiński Carpet Graph
• Index entries for linear recurrences with constant coefficients, signature (12,-35,24).

Crossrefs

Cf. A001018 (order), A271939 (size).

Cf. A365606 (degree 2), A365607 (degree 3), A365608 (degree 4).

Cf. A009964, A291066, A359452, A359453, A291066, A083233, A332705 (Menger sponge graph).

Programs

Mathematica

LinearRecurrence[{12,-35,24},{0,40,328},30] (* Paolo Xausa, Oct 16 2023 *)

Python

def A365607(n): return ((3<<3*n)+(3**(n-1)<<4))//5-8 # Chai Wah Wu, Nov 27 2023

Formula

a(n) = (3/5)*8^n + (16/15)*3^n - 8.

a(n) = 8*a(n-1) - 16*3^(n-2) + 56.

a(n) = 8^n - A365606(n) - A365608(n).

3*a(n) = 2*A271939(n) - 2*A365606(n) - 4*A365608(n).

G.f.: 8*x^2*(5 - 19*x)/((1 - x)*(1 - 3*x)*(1 - 8*x)). - Stefano Spezia, Sep 12 2023