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A098485 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square array such that their adjacency graph consists of a single component. Two positions (s,t), (u,v) are considered as adjacent if max(abs(s-u), abs(t-v)) <= 1.

%I A098485 #19 Jul 21 2024 11:36:16
%S A098485 1,4,6,9,20,48,16,42,132,419,25,72,256,973,3682,36,110,420,1747,7484,
%T A098485 31992,49,156,624,2741,12562,58620,273556,64,210,868,3955,18916,92912,
%U A098485 462104,2927505,81,272,1152,5389,26546,134868,697836,3644935,19082018
%N A098485 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square array such that their adjacency graph consists of a single component. Two positions (s,t), (u,v) are considered as adjacent if max(abs(s-u), abs(t-v)) <= 1.
%C A098485 Number of ways to mark the numbers on a square board on a lottery play slip such that one connected graphic pattern is formed. For the lottery "mark 6 numbers of 49 on a 7 X 7 grid of numbers" that is played in many countries, there are T(7,6)=58620 (out of binomial(49,6)=13983816) different combinations of 6 numbers whose graphic pattern on the board forms one connected component.
%H A098485 John Burkardt, <a href="http://www.csit.fsu.edu/~burkardt/f_src/grafpack/grafpack.html">GRAFPACK Graph Computations</a>.
%H A098485 Hugo Pfoertner, <a href="/A098485/a098485.txt">Counts of connected components in selected numbers on square lotto boards</a>.
%H A098485 Hugo Pfoertner, <a href="/A098485/a098485b.txt">Program to analyze the adjacency graph of selections on lotto boards</a>.
%e A098485 a(5)=T(3,2)=20 because there are 20 ways to mark two positions in a 3 X 3 square grid such that the two picked positions are either row-wise, column-wise or diagonally adjacent:
%e A098485 XX0...X00...X00...0XX...0X0...0X0...0X0...00X...00X...000
%e A098485 000...X00...0X0...000...X00...0X0...00X...0X0...00X...XX0
%e A098485 000...000...000...000...000...000...000...000...000...000
%e A098485 .........................................................
%e A098485 000...000...000...000...000...000...000...000...000...000
%e A098485 000...X00...0X0...000...X00...0X0...00X...0X0...00X...0XX
%e A098485 XX0...X00...X00...0XX...0X0...0X0...0X0...00X...00X...000
%o A098485 (Fortran) c See link.
%Y A098485 Cf. A090642, A098487 (selections where all marks are isolated from each other), A291716, A291717, A291718, A292152, A292153, A292154, A292155, A292156.
%K A098485 nonn,tabl
%O A098485 1,2
%A A098485 _Hugo Pfoertner_, Sep 14 2004