%I #16 Jun 27 2023 11:44:37

%S 2426,57152,1014458,16353152,253359866,3857162432,58255767098,

%T 876627759872,13168963989626,197671319438912,2966027888106938,

%U 44497125235352192,667503827640776186,10012886060527865792,150195591435759857978,2252949975250575898112

%N Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(4,n) (with n at least 4) missing three edges, where all three removed edges are incident to the same vertex in the 4-point set.

%C Start with a complete bipartite graph K(4,n) with vertex sets A and B where |A| = 4 and |B| is at least 4. 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 three edges, where all three removed edges are incident to the same point in A. So this sequence gives the number of sets at each location on the line segment between A' and B'.

%C Number of {0,1} 4 X n matrices (with n at least 4) with three fixed zero entries all in the same row and no zero rows or columns.

%C Take a complete bipartite graph K(4,n) (with n at least 4) having parts A and B where |A| = 4. This sequence gives the number of edge covers of the graph obtained from this K(4,n) graph after removing three edges, where all three removed edges are incident 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_04">Index entries for linear recurrences with constant coefficients</a>, signature (26,-196,486,-315).

%F a(n) = 343*15^(n-3) - 424*7^(n-3) + 28*3^(n-2) - 3.

%F From _Stefano Spezia_, Jan 26 2021: (Start)

%F G.f.: 2*x^4*(1213 - 2962*x + 2001*x^2)/(1 - 26*x + 196*x^2 - 486*x^3 + 315*x^4).

%F a(n) = 26*a(n-1) - 196*a(n-2) + 486*a(n-3) - 315*a(n-4) for n > 7. (End)

%Y Other sequences of segments from removing edges from bipartite graphs: A335608-A335613, A337416-A337418.

%Y Polygonal chain sequences: A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939.

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

%K easy,nonn

%O 4,1

%A _Roman I. Vasquez_, Jan 25 2021