A342850 - OEIS

A342850

Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(3,n) (with n at least 4) missing three edges, where all three removed edges are incident to different vertices in the 3-point set and none of the removed edges are incident to the same vertex in the other set.

162, 1242, 9018, 64098, 451602, 3169962, 22215978, 155590578, 1089370242, 7626300282, 53386227738, 373709971458, 2615988932082, 18311979920202, 128184031628298, 897288737958738, 6281022715393122, 43967163656797722, 307770159544721658, 2154391158654464418, 15080738236105489362

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OFFSET

4,1

COMMENTS

Start with a complete bipartite graph K(3,n) with vertex sets A and B where |A| = 3 and |B| is at least 4. We can arrange the points in sets A and B such that h(A,B) = d(a,b) for all a in A and b in B, where h is the Hausdorff metric. The pair [A,B] is a configuration. Then a set C is between A and B at location s if h(A,C) = h(C,B) = h(A,B) and h(A,C) = s. Call a pair ab, where a is in A and b is in B an edge. This sequence provides the number of sets between sets A' and B' at location s in a new configuration [A',B'] obtained from [A,B] by removing three edges, where all three removed edges are incident to different points in A and none of the removed edges are incident to the same point in B. So this sequence tells the number of sets at each location on the line segment between A' and B'.

Number of {0,1} 3 X n matrices (with n at least 4) with three fixed zero entries none of which are in the same row or column with no zero rows or columns.

Take a complete bipartite graph K(3,n) (with n at least 4) having parts A and B where |A| = 3. This sequence gives the number of edge covers of the graph obtained from this K(3,n) graph after removing three edges, where all three removed edges are incident to different vertices in A and none of the removed edges are incident to the same vertex in B.

LINKS

Paolo Xausa, Table of n, a(n) for n = 4..1000

Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.

Index entries for linear recurrences with constant coefficients, signature (10,-21).

FORMULA

a(n) = 27*7^(n-3) - 3^(n-1).

G.f.: 54*x^4*(3 - 7*x)/(1 - 10*x + 21*x^2). - Stefano Spezia, Mar 25 2021

MATHEMATICA

A342850[n_] := 27*7^(n-3) - 3^(n-1); Array[A342850, 25, 4] (* Paolo Xausa, Jul 01 2026 *)

CROSSREFS