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.

2792, 140114, 5366288, 183405386, 5953824632, 188681559554, 5911452093728, 184194287464826, 5724142958302472, 177660449252559794, 5510655708296433968, 170878064308411409066, 5297936128237164553112, 164246762516365548788834, 5091810779768636860563008

Offset: 3

Author

Steven Schlicker, Dec 30 2020

Keywords

• easy
• nonn

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.
• Index entries for linear recurrences with constant coefficients, signature (57,-1002,6562,-15381,9765).

Crossrefs

Sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418, A340173-A340175, A340199-A340201, A340897-A340899, A342580, A342796, A342850, A340403-A340405, A340433-A340438, A341551-A341553, A342327-A342328, A343372-A343374, A343800.

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).

Programs

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 *)

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)