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A365608
Number of degree 4 vertices in the n-Sierpinski carpet graph.
9
0, 4, 100, 1060, 9316, 77092, 624484, 5019172, 40223332, 321996580, 2576602468, 20614709284, 164923342948, 1319403749668, 10555281015652, 84442401180196, 675539668606564, 5404318726347556, 43234553943265636, 345876443943580708, 2767011588741012580, 22136092821505201444, 177088742906772914020
OFFSET
1,2
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, SierpiƄski Carpet Graph
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
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, 4, 100}, 30] (* Paolo Xausa, Oct 16 2023 *)
PROG
(Python)
def A365608(n): return ((3<<3*n-1)-(3**(n-1)<<5))//5+4 # 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: A158082 A017090 A029995 * A244352 A173987 A052144
KEYWORD
nonn,easy
AUTHOR
Allan Bickle, Sep 12 2023
STATUS
approved