A337416 Number of sets (in the Hausdorff metric geometry) 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 removed edges are incident to the same point in the 5 point part.

2240, 133232, 5366288, 187074656, 6126049760, 194922245072, 6118612137008, 190822947290816, 5932740419114240, 184173665371614512, 5713266248795701328, 177169506604462719776, 5493128593023515417120, 170300095372377973419152, 5279499596024093537691248

Offset: 3

Author

Steven Schlicker, Aug 26 2020

Keywords

• easy
• nonn

Comments

The Hausdorff metric defines a distance between sets. Using this distance we can define line segments with sets as endpoints. Create two sets from the vertices of the parts A and B (with |A| = 5) of a complete bipartite graph K(5,n) (with n at least 3) missing two edges, where the removed edges are incident to the same point in A. Points in the sets A and B that correspond to vertices that are connected by edges are the same Euclidean distance apart. 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 (with n at least 3) matrices with two fixed zero entries in the same row 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 incident to the same vertex in A.

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. Number of {0,1} n X n matrices with no zero rows or columns A048291.

Programs

Maple

a:= proc(n) 225*31^(n-2) - 421*15^(n-2)+250*7^(n-2)-58*3^(n-2)+4 end proc: seq(a(n), n=3..20);

Formula

a(n) = 225*31^(n-2) - 421*15^(n-2) + 250*7^(n-2) - 58*3^(n-2) + 4.

From Colin Barker, Oct 13 2020: (Start)

G.f.: 16*x^3*(140 + 347*x + 1034*x^2 - 261*x^3) / ((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) for n>7.

(End)