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A365607 Number of degree 3 vertices in the n-Sierpinski carpet graph. 8
0, 40, 328, 2536, 19912, 158056, 1260616, 10073320, 80551624, 644308072, 5154149704, 41232252904, 329855188936, 2638833008488, 21110638558792, 168885031942888, 1351080025960648, 10808639518937704, 86469114085259080, 691752906483344872, 5534023233270575560, 44272185810376054120 (list; graph; refs; listen; history; text; internal format)
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
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
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
KEYWORD
nonn,easy
AUTHOR
Allan Bickle, Sep 12 2023
STATUS
approved

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Last modified March 4 05:23 EST 2024. Contains 370522 sequences. (Running on oeis4.)