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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.
39

%I #20 Jun 27 2023 11:11:30

%S 2240,133232,5366288,187074656,6126049760,194922245072,6118612137008,

%T 190822947290816,5932740419114240,184173665371614512,

%U 5713266248795701328,177169506604462719776,5493128593023515417120,170300095372377973419152,5279499596024093537691248

%N 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.

%C 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.

%C 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.

%C 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.

%H Steven Schlicker, Roman Vasquez, and Rachel Wofford, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Wofford/wofford4.html">Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (57,-1002,6562,-15381,9765).

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

%F From _Colin Barker_, Oct 13 2020: (Start)

%F 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)).

%F 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.

%F (End)

%p 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);

%Y 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.

%K easy,nonn

%O 3,1

%A _Steven Schlicker_, Aug 26 2020