cp's OEIS Frontend

A384499 Number of paths from the top to bottom of a 3-dimensional triangular bipyramidal graph of height 2n, with no repeated vertices, and no upward moves.

%I A384499 #42 Jun 18 2025 16:41:12
%S A384499 1,15,11475,1093007025,52244816853213675,6472823166678668309527843125,
%T A384499 11561557982049161046080105648122197757331625,
%U A384499 1687343403738428640604090554388660433120115565168405371811095975
%N A384499 Number of paths from the top to bottom of a 3-dimensional triangular bipyramidal graph of height 2n, with no repeated vertices, and no upward moves.
%C A384499 Let T(k) be a triangular grid graph as defined in Weisstein. T(k) has k*(k+1)/2 aligned triangles that are pointing the same way as T(k) and k*(k-1)/2 triangles pointing the opposite way.
%C A384499 The triangular bipyramidal graph TBP(n) can be constructed as follows:
%C A384499 - Instantiate the layers of the graph from top to bottom T(0) ... T(n-1) T(n) T(n-1) ...  T(0).
%C A384499 - Align the (k+1)*(k+2)/2 vertices in layer T(k) with the (k+1)*(k+2)/2 aligned triangles in adjacent layer T(k+1) to create a 1:1 correspondence between vertices in T(k) and aligned triangles in T(k+1)
%C A384499 - Connect each vertex in T(k) with the 3 vertices of the corresponding aligned triangle in the adjacent T(k+1).
%C A384499 Equivalently, extending the logic from Weisstein, the triangular bipyramidal graph TBP(n) is the graph on vertices (i,j,k,l) with 0 <= i,j,k <= n, -n <= l <= n, and i+j+k+|l| == n such that vertices are adjacent if the sum of absolute differences of the coordinates of two vertices is 2, and the absolute difference of the l coordinate is 0 or 1. That is, vertices are adjacent if |i1-i2| + |j1-j2| + |k1-k2| + |l1-l2| == 2 and |l1-l2| <= 1.
%H A384499 Sameer Gauria, <a href="https://github.com/sgauria/Fiddler/blob/main/Fiddler_2025_05_16%20%20-%20Summary%20for%20OEIS.ipynb">Fiddler_2025_05_16 - Summary for OEIS</a>
%H A384499 Tom Keith, <a href="https://bsky.app/profile/tomkeith.bsky.social/post/3lpon2szb5s2s">Bluesky post.</a>
%H A384499 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGridGraph.html">Triangular Grid Graph</a>
%H A384499 Zach Wissner-Gross, <a href="https://thefiddler.substack.com/p/can-you-permeate-the-pyramid">Can You Permeate the Pyramid?</a>
%H A384499 Zach Wissner-Gross, <a href="https://thefiddler.substack.com/p/how-long-is-the-river-of-text">How Long Is the River?</a>
%e A384499 a(1)=15: TBP(1) has 5 vertices A, B, C, D, E. Vertex A (top) is connected to vertices B, C, D. B, C, D are connected to each other. B, C, D are connected to E (bottom). The only valid paths are: ABE, ACE, ADE, ABCE, ABDE, ACBE, ACDE, ADBE, ADCE, ABCDE, ABDCE, ACBDE, ACDBE, ADBCE, ADCBE. For instance, path ABCADE is not valid because of upward move (CA) and repeated vertex (A).
%Y A384499 Cf. A002454 (2-dimensional version).
%K A384499 nonn,walk
%O A384499 0,2
%A A384499 _Sameer Gauria_, May 31 2025