A289236 Square array a(p,q) read by antidiagonals: a(p,q) = the number of line segments that constitute the trajectory of a billiard ball on a pool table with dimensions p X q, before the ball reaches a corner.

1, 2, 2, 3, 1, 3, 4, 4, 4, 4, 5, 2, 1, 2, 5, 6, 6, 6, 6, 6, 6, 7, 3, 7, 1, 7, 3, 7, 8, 8, 2, 8, 8, 2, 8, 8, 9, 4, 9, 4, 1, 4, 9, 4, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 5, 3, 2, 11, 1, 11, 2, 3, 5, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

Author

Luc Rousseau, Jun 28 2017

Keywords

• nonn
• tabl

Comments

The billiard game considered here is an idealized one: the pool table is a rectangle with vertices (0,0), (p,0), (p,q), (0, q); the ball is shrunk to a point and is launched from vertex (0,0) with initial velocity vector (1,1); collisions are supposed elastic and friction is supposed nonexistent, so that the ball can never stop on the table; when the ball bounces, the angle of reflection is equal to the angle of incidence; the ball can only exit through a vertex.

a(p,q) counts the line segments that constitute the trajectory.

Examples

			In a square-shaped pool table, the ball just crosses diagonally. a(p,p)=1.
In a pool table of dimensions 2 X 1, the ball bounces once and exits. a(2,1)=2.
The square array a(p,q) begins:
   1  2  3  4  5  6  7
   2  1  4  2  6  3  8
   3  4  1  6  7  2  9
   4  2  6  1  8  4 10
   5  6  7  8  1 10 11
   6  3  2  4 10  1 12
   7  8  9 10 11 12  1
		

Crossrefs

Cf. A059026 (the triangle version).

Programs

Java

long a(long p, long q) {
long i = 0, x = 0, y = 0, dx = +1, dy = +1, s = 1;
while ((((x % p) != 0) || ((y % q) != 0)) || (i == 0)) {
i ++; long xx = x + dx; long yy = y + dy;
boolean xok = (0 <= xx) && (xx <= p);
boolean yok = (0 <= yy) && (yy <= q);
if (xok && yok) { x = xx; y = yy; }
else { s ++;
if (! xok) { dx = -dx; }
if (! yok) { dy = -dy; }
}} return s; }

Formula

a(p,q) = (p + q) / gcd(p, q) - 1.