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%I #19 Aug 17 2024 23:08:22
%S 6,15,42,132,456,1680,6432,25152,99456,395520,1577472,6300672,
%T 25184256,100700160,402726912,1610760192,6442745856,25770393600,
%U 103080394752,412319219712,1649272160256,6597079203840,26388297940992,105553154015232,422212540563456,1688850011258880,6755399743045632
%N Number of pairs of antipodal vertices in the level n>1 Sierpiński triangle graph.
%C A level 1 Sierpiński triangle graph is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles.
%C Antipodal vertices are a pair of vertices at maximum distance in a graph. The diameter of the level n Sierpiński triangle graph is 2^(n-1).
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).
%H Allan Bickle, <a href="https://allanbickle.wordpress.com/wp-content/uploads/2016/05/sierpinskigraphpaper2.pdf">Properties of Sierpinski Triangle Graphs</a>, Springer PROMS 448 (2021) 295-303.
%H A. Hinz, S. Klavzar, and S. Zemljic, <a href="https://doi.org/10.1016/j.dam.2016.09.024">A survey and classification of Sierpinski-type graphs</a>, Discrete Applied Mathematics 217 3 (2017), 565-600.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiGasketGraph.html">Sierpiński Gasket Graph</a>
%F a(n) = 3*2^(n-3)*(2^(n-2)+3).
%F a(n) = 3*A257273(n-2).
%F a(n) = A375256(n-1) + 3.
%e 3 example graphs: o
%e / \
%e o---o
%e / \ / \
%e o o---o---o
%e / \ / \ / \
%e o o---o o---o o---o
%e / \ / \ / \ / \ / \ / \ / \
%e o---o o---o---o o---o---o---o---o
%e Graph: S_1 S_2 S_3
%e For S_2, there are 3 pairs of corners and 3 pairs of a corner and a middle vertex, so a(2) = 6.
%o (PARI) a(n) = 3*2^(n-3)*(2^(n-2)+3); \\ _Michel Marcus_, Aug 08 2024
%Y Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959, A298202 (Sierpiński triangle graphs).
%Y Cf. A375256 (antipodal pairs in Hanoi graphs).
%K nonn,easy
%O 2,1
%A _Allan Bickle_, Aug 07 2024
%E More terms from _Michel Marcus_, Aug 08 2024
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