A370933 Number of pairs of antipodal vertices in the level n>1 Sierpiński triangle graph.

6, 15, 42, 132, 456, 1680, 6432, 25152, 99456, 395520, 1577472, 6300672, 25184256, 100700160, 402726912, 1610760192, 6442745856, 25770393600, 103080394752, 412319219712, 1649272160256, 6597079203840, 26388297940992, 105553154015232, 422212540563456, 1688850011258880, 6755399743045632

Offset: 2

Author

Allan Bickle, Aug 07 2024

Keywords

• nonn
• easy

Comments

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.

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).

Examples

			3 example graphs:                        o
                                        / \
                                       o---o
                                      / \ / \
                        o            o---o---o
                       / \          / \     / \
             o        o---o        o---o   o---o
            / \      / \ / \      / \ / \ / \ / \
           o---o    o---o---o    o---o---o---o---o
Graph:      S_1        S_2              S_3
For S_2, there are 3 pairs of corners and 3 pairs of a corner and a middle vertex, so a(2) = 6.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-8).
• Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
• A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
• Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph

Crossrefs

Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959, A298202 (Sierpiński triangle graphs).

Cf. A375256 (antipodal pairs in Hanoi graphs).

Programs

Mathematica

A370933[n_] := 3*2^(n - 3)*(2^(n - 2) + 3);
Array[A370933, 30, 2] (* or *)
LinearRecurrence[{6, -8}, {6, 15}, 30] (* Paolo Xausa, Sep 23 2024 *)

PARI

a(n) = 3*2^(n-3)*(2^(n-2)+3); \\ Michel Marcus, Aug 08 2024

Formula

a(n) = 3*2^(n-3)*(2^(n-2)+3).

a(n) = 3*A257273(n-2).

a(n) = A375256(n-1) + 3.

Extensions

More terms from Michel Marcus, Aug 08 2024