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A340201
Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(5,n) (with n at least 3) missing two edges, where the two removed edges are not incident to the same vertex in the 5-point set and are also not incident to the same vertex in the other set.
36
2899, 145387, 5566147, 190200379, 6173845939, 195645606667, 6129507633187, 190986695659099, 5935198857377299, 184210557438511147, 5713819738261143427, 177177809705712311419, 5493253144857237049459, 170301963687088948318027, 5279527621005195132400867
OFFSET
3,1
COMMENTS
Start with a complete bipartite graph K(5,n) with vertex sets A and B where |A| = 5 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} 5 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(5,n) (with n at least 3) having parts A and B where |A| = 5. This sequence gives the number of edge covers of the graph obtained from this K(5,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.
FORMULA
a(n) = 225*31^(n-2) - 357*15^(n-2) + 202*7^(n-2) - 46*3^(n-2) + 3.
From Alejandro J. Becerra Jr., Feb 14 2021: (Start)
G.f.: x^3*(263655*x^4 - 415464*x^3 + 183886*x^2 - 19856*x + 2899)/((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)*(1 - 31*x)).
a(n) = 57*a(n-1) - 1002*a(n-2) + 6562*a(n-3) - 15381*a(a-4) + 9765*a(n-5). (End)
CROSSREFS
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.
Sequence in context: A285756 A248109 A219963 * A235102 A252267 A207281
KEYWORD
easy,nonn
AUTHOR
Roman I. Vasquez, Dec 31 2020
STATUS
approved