1,2

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.

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

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

(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; }

Cf. A059026 (the triangle version).

Sequence in context: A191384 A191305 A227287 * A280172 A337942 A341458

Adjacent sequences: A289233 A289234 A289235 * A289237 A289238 A289239

nonn,tabl

Luc Rousseau, Jun 28 2017

approved