A367702 - OEIS

%I #18 Nov 29 2023 07:03:13

%S 0,144,2784,57552,1180320,23889936,480221280,9624275280,192645717024,

%T 3854200280208,77094305873376,1541968557881808,30840030795738528,

%U 616805893363960080,12336160087905835872,246723539526229152336,4934473492678780614432,98689491470837087102352

%N Number of degree 4 vertices in the n-Menger sponge graph.

%C The level 0 Menger sponge graph is a single vertex.  The level n Menger sponge graph is formed from 20 copies of level n-1 in the shape of a cube with middle faces removed by joining boundary vertices between adjacent copies.

%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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (32,-275,724,-480).

%F a(n) = (32/85)*20^n - (4/5)*8^n + (648/85)*3^n - 24.

%F a(n) = 20*a(n-1) + (6/5)*8^n - (216/5)*3^n + 456.

%F a(n) = 20^n - A367700(n) - A367701(n) - A367706(n) - A367707(n).

%F 4*a(n) = 2*A291066(n) - 2*A367700(n) - 3*A367701(n) - 5*A367706(n) - 6*A367707(n).

%F G.f.: 12*x^2*(7 - 224*x + 1865*x^2 - 4308*x^3)/(5*(1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - _Stefano Spezia_, Nov 28 2023

%e The level 1 Menger sponge graph is a cube with each edge subdivided, which has 12 degree 2 vertices and 8 degree 3 vertices.  Thus a(1) = 0.

%t LinearRecurrence[{32,-275,724,-480},{0,144,2784,57552},25] (* _Paolo Xausa_, Nov 29 2023 *)

%o (Python)

%o def A367702(n): return ((5**n<<(n<<1)+5)-(17<<(3*n+2))+(3**(n+4)<<3))//85-24 # _Chai Wah Wu_, Nov 28 2023

%Y Cf. A009964 (number of vertices), A291066 (number of edges).

%Y Cf. A359452, A359453 (numbers of corner and non-corner vertices).

%Y Cf. A291066, A083233, A332705 (surface area).

%Y Cf. A367700, A367701, A367702, A367706, A367707 (degrees 2 through 6).

%Y Cf. A001018, A271939, A365606, A365607, A365608 (Sierpinski carpet graphs).

%K nonn,easy

%O 1,2

%A _Allan Bickle_, Nov 27 2023