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.

1, 4, 6, 9, 20, 48, 16, 42, 132, 419, 25, 72, 256, 973, 3682, 36, 110, 420, 1747, 7484, 31992, 49, 156, 624, 2741, 12562, 58620, 273556, 64, 210, 868, 3955, 18916, 92912, 462104, 2927505, 81, 272, 1152, 5389, 26546, 134868, 697836, 3644935, 19082018

Offset: 1

Author

Hugo Pfoertner, Sep 14 2004

Keywords

• nonn
• tabl

Comments

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.

Examples

			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:
XX0...X00...X00...0XX...0X0...0X0...0X0...00X...00X...000
000...X00...0X0...000...X00...0X0...00X...0X0...00X...XX0
000...000...000...000...000...000...000...000...000...000
.........................................................
000...000...000...000...000...000...000...000...000...000
000...X00...0X0...000...X00...0X0...00X...0X0...00X...0XX
XX0...X00...X00...0XX...0X0...0X0...0X0...00X...00X...000
		

Links

• John Burkardt, GRAFPACK Graph Computations.
• Hugo Pfoertner, Counts of connected components in selected numbers on square lotto boards.
• Hugo Pfoertner, Program to analyze the adjacency graph of selections on lotto boards.

Crossrefs

Cf. A090642, A098487 (selections where all marks are isolated from each other), A291716, A291717, A291718, A292152, A292153, A292154, A292155, A292156.

Programs

Fortran

c See link.