A335611 - OEIS

%I #32 Jun 27 2023 11:11:25

%S 242,49208,5049626,397551920,27839280002,1845793079528,

%T 119216755050026,7602793781214560,481851209165874962,

%U 30446042035976733848,1920876815510991751226,121101364739596962016400,7632056827800217741372322,480902390923479550619876168

%N Number of sets (in the Hausdorff metric geometry) at each location between two sets defined by a complete bipartite graph K(6,n) (with n at least 2) missing one edge.

%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 of a complete bipartite graph K(6,n) (with n at least 2) missing one edge so that 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 A and B.

%C Number of {0,1} 6 X n (with n at least 2) matrices with one fixed zero entry and no zero rows or columns.

%C Take a complete bipartite graph K(6,n) (with n at least 2). This sequence gives the number of edge covers of the graph obtained from this K(6,n) graph after removing one edge.

%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_06">Index entries for linear recurrences with constant coefficients</a>, signature (120,-4593,69688,-428787,978768,-615195).

%F a(n) = 31*63^(n-1) - 106*31^(n-1) + 145*15^(n-1) - 100*7^(n-1) + 35*3^(n-1) - 5.

%F From _Colin Barker_, Jul 17 2020: (Start)

%F G.f.: 2*x^2*(121 + 10084*x + 128086*x^2 + 372324*x^3 + 270585*x^4) / ((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)*(1 - 31*x)*(1 - 63*x)).

%F a(n) = 120*a(n-1) - 4593*a(n-2) + 69688*a(n-3) - 428787*a(n-4) + 978768*a(n-5) - 615195*a(n-6) for n>7.

%F (End)

%p a:= proc(n) 31*63^(n-1)-106*31^(n-1)+145*15^(n-1) - 100*7^(n-1)+35*3^(n-1)-5 end proc: seq(a(n), n=2..20);

%o (PARI) Vec(2*x^2*(121 + 10084*x + 128086*x^2 + 372324*x^3 + 270585*x^4) / ((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)*(1 - 31*x)*(1 - 63*x)) + O(x^18)) \\ _Colin Barker_, Jul 17 2020

%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 2,1

%A _Steven Schlicker_, Jul 16 2020