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A335612
Number of sets (in the Hausdorff metric geometry) at each location between two sets defined by a complete bipartite graph K(3,n) (with n at least 3) missing two edges, where the removed edges are incident to the same vertex in the three point part.
0
32, 344, 2792, 20720, 148592, 1050824, 7387832, 51811040, 362965952, 2541627704, 17793992072, 124565738960, 871983556112, 6103955042984, 42727895751512, 299095901612480, 2093673205343072, 14655718119568664, 102590043883482152
OFFSET
3,1
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| = 3) of a complete bipartite graph K(3,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 gives the number of sets at each location on the line segment between the sets A and B.
Number of {0,1} 3 X n matrices (with n at least 3) with two fixed zero entries in the same row and no zero rows or columns.
Take a complete bipartite graph K(3,n) (with n at least 3) 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 two edges, where the two removed edges are incident to the same vertex in A.
FORMULA
a(n) = 9*7^(n-2) - 11*3^(n-2) + 2.
From Colin Barker, Jul 17 2020: (Start)
G.f.: 8*x^3*(4 - x) / ((1 - x)*(1 - 3*x)*(1 - 7*x)).
a(n) = 11*a(n-1) - 31*a(n-2) + 21*a(n-3) for n>5.
(End)
MAPLE
a:= proc(n) 9*7^(n-2)-11*3^(n-2)+2 end proc: seq(a(n), n=3..21);
PROG
(PARI) Vec(8*x^3*(4 - x) / ((1 - x)*(1 - 3*x)*(1 - 7*x)) + O(x^25)) \\ Colin Barker, Jul 17 2020
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: A009797 A240293 A254953 * A006763 A158074 A200531
KEYWORD
easy,nonn
AUTHOR
Steven Schlicker, Jul 16 2020
STATUS
approved