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A340174 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 but are incident to the same vertex in the other set. 0
2792, 140114, 5366288, 183405386, 5953824632, 188681559554, 5911452093728, 184194287464826, 5724142958302472, 177660449252559794, 5510655708296433968, 170878064308411409066, 5297936128237164553112, 164246762516365548788834, 5091810779768636860563008 (list; graph; refs; listen; history; text; internal format)
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 but 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} 5 X n matrices (with n at least 3) with two fixed zero entries in the same 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 but are 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) = 7*31^(n-1) - 23*15^(n-1) + 4*7^n - 5*3^(n) + 3.
From Alejandro J. Becerra Jr., Feb 12 2021: (Start)
G.f.: 2*x^3*(126945*x^4 - 199953*x^3 + 88687*x^2 - 9515*x + 1396)/((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(n-4) + 9765*a(n-5). (End)
MATHEMATICA
Array[7*31^(# - 1) - 23*15^(# - 1) + 4*7^# - 5*3^(#) + 3 &, 15, 3] (* Michael De Vlieger, Jan 12 2021 *)
LinearRecurrence[{57, -1002, 6562, -15381, 9765}, {2792, 140114, 5366288, 183405386, 5953824632}, 20] (* Harvey P. Dale, Aug 11 2021 *)
CROSSREFS
Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939.
Cf. A048291 (number of {0,1} n X n matrices with no zero rows or columns).
Sequence in context: A106300 A252601 A233923 * A252900 A236637 A236943
KEYWORD
easy,nonn
AUTHOR
Steven Schlicker, Dec 30 2020
STATUS
approved

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Last modified April 16 07:08 EDT 2024. Contains 371698 sequences. (Running on oeis4.)