A365608 Number of degree 4 vertices in the n-Sierpinski carpet graph.

0, 4, 100, 1060, 9316, 77092, 624484, 5019172, 40223332, 321996580, 2576602468, 20614709284, 164923342948, 1319403749668, 10555281015652, 84442401180196, 675539668606564, 5404318726347556, 43234553943265636, 345876443943580708, 2767011588741012580, 22136092821505201444, 177088742906772914020

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},{0,4,100},30] (* Paolo Xausa, Oct 16 2023 *)

Python

def A365608(n): return ((3<<3*n-1)-(3**(n-1)<<5))//5+4 # Chai Wah Wu, Nov 27 2023

Formula

a(n) = (3/10)*8^n - (32/15)*3^n + 4.

a(n) = 8*a(n-1) + 32*3^(n-2) - 28.

a(n) = 8^n - A365606(n) - A365607(n).

4*a(n) = 2*A271939(n) - 2*A365606(n) - 3*A365607(n).

G.f.: 4*x^2*(1 + 13*x)/((1 - x)*(1 - 3*x)*(1 - 8*x)). - Stefano Spezia, Sep 12 2023