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

1,2

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.

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).

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

Example

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.

Mathematica

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

Prog

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

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).

Sequence in context: A061993 A087954 A160328 * A247407 A251431 A285919

Adjacent sequences: A365604 A365605 A365606 * A365608 A365609 A365610

Keyword

nonn,easy

Author

Allan Bickle, Sep 12 2023

Status

approved