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A343372
Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(3,n) (with n at least 4) missing three edges, where exactly two removed edges are incident to the same vertex in the 3-point set and exactly two removed edges are incident to the same vertex in the other set.
36
112, 922, 6880, 49450, 350032, 2461882, 17268160, 120982090, 847189552, 5931271642, 41521735840, 290660653930, 2034650086672, 14242627134202, 99698619521920, 697891025400970, 4885239244049392
OFFSET
4,1
COMMENTS
Start with a complete bipartite graph K(3,n) with vertex sets A and B where |A| = 3 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 exactly two removed edges are incident to the same point in A and exactly two removed edges 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} 3 X n matrices (with n at least 4) with three fixed zero entries where exactly two zero entries occur in one row and exactly two zero entries occur in one column, with no zero rows or columns.
Take a complete bipartite graph K(3,n) (with n at least 4) having parts A and B where |A| = 3. This sequence gives the number of edge covers of the graph obtained from this K(3,n) graph after removing three edges, where exactly two removed edges are incident to the same vertex in A and exactly two removed edges are incident to the same vertex in B.
FORMULA
a(n) = 3*7^(n-2) - 4*3^(n-2) + 1.
G.f.: 2*x^4*(56 - 155*x + 105*x^2)/(1 - 11*x + 31*x^2 - 21*x^3). - Stefano Spezia, Apr 13 2021
MATHEMATICA
Drop[CoefficientList[Series[2 x^4*(56 - 155 x + 105 x^2)/(1 - 11 x + 31 x^2 - 21 x^3), {x, 0, 20}], x], 4] (* Michael De Vlieger, Apr 13 2021 *)
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.
Sequence in context: A211448 A206318 A206311 * A330418 A235311 A008430
KEYWORD
easy,nonn
AUTHOR
Steven Schlicker, Apr 12 2021
STATUS
approved