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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.

%I A289236 #20 Aug 12 2018 17:28:07
%S A289236 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,
%T A289236 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,
%U A289236 2,3,5,11,12,12,12,12,12,12,12,12,12,12,12,12
%N 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.
%C A289236 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.
%C A289236 a(p,q) counts the line segments that constitute the trajectory.
%F A289236 a(p,q) = (p + q) / gcd(p, q) - 1.
%e A289236 In a square-shaped pool table, the ball just crosses diagonally. a(p,p)=1.
%e A289236 In a pool table of dimensions 2 X 1, the ball bounces once and exits. a(2,1)=2.
%e A289236 The square array a(p,q) begins:
%e A289236    1  2  3  4  5  6  7
%e A289236    2  1  4  2  6  3  8
%e A289236    3  4  1  6  7  2  9
%e A289236    4  2  6  1  8  4 10
%e A289236    5  6  7  8  1 10 11
%e A289236    6  3  2  4 10  1 12
%e A289236    7  8  9 10 11 12  1
%o A289236 (Java)
%o A289236 long a(long p, long q) {
%o A289236 long i = 0, x = 0, y = 0, dx = +1, dy = +1, s = 1;
%o A289236 while ((((x % p) != 0) || ((y % q) != 0)) || (i == 0)) {
%o A289236 i ++; long xx = x + dx; long yy = y + dy;
%o A289236 boolean xok = (0 <= xx) && (xx <= p);
%o A289236 boolean yok = (0 <= yy) && (yy <= q);
%o A289236 if (xok && yok) { x = xx; y = yy; }
%o A289236 else { s ++;
%o A289236 if (! xok) { dx = -dx; }
%o A289236 if (! yok) { dy = -dy; }
%o A289236 }} return s; }
%Y A289236 Cf. A059026 (the triangle version).
%K A289236 nonn,tabl
%O A289236 1,2
%A A289236 _Luc Rousseau_, Jun 28 2017