A365607 - OEIS

%I #24 Feb 16 2025 08:34:06

%S 0,40,328,2536,19912,158056,1260616,10073320,80551624,644308072,

%T 5154149704,41232252904,329855188936,2638833008488,21110638558792,

%U 168885031942888,1351080025960648,10808639518937704,86469114085259080,691752906483344872,5534023233270575560,44272185810376054120

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

%C 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.

%H Paolo Xausa, <a href="/A365607/b365607.txt">Table of n, a(n) for n = 1..1000</a>

%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/mengerspongedegree.pdf">Degrees of Menger and Sierpinski Graphs</a>, Congr. Num. 227 (2016) 197-208.

%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/mengerspongeshort.pdf">MegaMenger Graphs</a>, The College Mathematics Journal, 49 1 (2018) 20-26.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiCarpetGraph.html">Sierpiński Carpet Graph</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-35,24).

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

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

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

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

%F G.f.: 8*x^2*(5 - 19*x)/((1 - x)*(1 - 3*x)*(1 - 8*x)). - _Stefano Spezia_, Sep 12 2023

%e 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.

%t LinearRecurrence[{12,-35,24},{0,40,328},30] (* _Paolo Xausa_, Oct 16 2023 *)

%o (Python)

%o def A365607(n): return ((3<<3*n)+(3**(n-1)<<4))//5-8 # _Chai Wah Wu_, Nov 27 2023

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

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

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

%K nonn,easy,changed

%O 1,2

%A _Allan Bickle_, Sep 12 2023