A340199 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 3) missing two edges, where the two removed edges are not incident to the same vertex in the 3-point set and are also not incident to the same vertex in the other set.

43, 379, 2899, 21043, 149563, 1053739, 7396579, 51837283, 363044683, 2541863899, 17794700659, 124567864723, 871989933403, 6103974174859, 42727953147139, 299096073799363, 2093673721903723, 14655719669250619, 102590048532528019

Offset: 3

Author

Roman I. Vasquez, Dec 31 2020

Keywords

• easy
• nonn

Comments

Start with a complete bipartite graph K(3,n) with vertex sets A and B where |A| = 3 and |B| is at least 3. 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 two edges, where the two removed edges are not incident to the same point in A and are also not 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 3) with two fixed zero entries not in the same row or column and no zero rows or columns.

Take a complete bipartite graph K(3,n) (with n at least 3) 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 two edges, where the two removed edges are not incident to the same vertex in A and are also not incident to the same vertex in B.

Links

• 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 (11,-31,21).

Crossrefs

Other sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418. Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939. Number of {0,1} n X n matrices with no zero rows or columns A048291.

Programs

Mathematica

LinearRecurrence[{11,-31,21},{43,379,2899},20] (* Harvey P. Dale, Apr 10 2024 *)

Formula

a(n) = 9*7^(n-2) - 7*3^(n-2) + 1.

From Stefano Spezia, Dec 31 2020: (Start)

G.f.: x^3*(43 - 94*x + 63*x^2)/(1 - 11*x + 31*x^2 - 21*x^3).

a(n) = 11*a(n-1) - 31*a(n-2) + 21*a(n-3) for n > 5. (End)