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.

1, 15, 11475, 1093007025, 52244816853213675, 6472823166678668309527843125, 11561557982049161046080105648122197757331625, 1687343403738428640604090554388660433120115565168405371811095975

Offset: 0

Author

Sameer Gauria, May 31 2025

Keywords

• nonn
• walk

Comments

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.

The triangular bipyramidal graph TBP(n) can be constructed as follows:

- Instantiate the layers of the graph from top to bottom T(0) ... T(n-1) T(n) T(n-1) ... T(0).

- 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)

- Connect each vertex in T(k) with the 3 vertices of the corresponding aligned triangle in the adjacent T(k+1).

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.

Examples

			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).
		

Links

• Sameer Gauria, Fiddler_2025_05_16 - Summary for OEIS
• Tom Keith, Bluesky post.
• Eric Weisstein's World of Mathematics, Triangular Grid Graph
• Zach Wissner-Gross, Can You Permeate the Pyramid?
• Zach Wissner-Gross, How Long Is the River?

Crossrefs

Cf. A002454 (2-dimensional version).