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%I #22 Nov 27 2023 19:26:47
%S 0,4,100,1060,9316,77092,624484,5019172,40223332,321996580,2576602468,
%T 20614709284,164923342948,1319403749668,10555281015652,84442401180196,
%U 675539668606564,5404318726347556,43234553943265636,345876443943580708,2767011588741012580,22136092821505201444,177088742906772914020
%N Number of degree 4 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="/A365608/b365608.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="http://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) = (3/10)*8^n - (32/15)*3^n + 4.
%F a(n) = 8*a(n-1) + 32*3^(n-2) - 28.
%F a(n) = 8^n - A365606(n) - A365607(n).
%F 4*a(n) = 2*A271939(n) - 2*A365606(n) - 3*A365607(n).
%F G.f.: 4*x^2*(1 + 13*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},{0,4,100},30] (* _Paolo Xausa_, Oct 16 2023 *)
%o (Python)
%o def A365608(n): return ((3<<3*n-1)-(3**(n-1)<<5))//5+4 # _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
%O 1,2
%A _Allan Bickle_, Sep 12 2023
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