A099407 Number of points of self-intersection of the path of a billiard ball traveling at a 45-degree angle on a prime(n) X prime(n+1) billiard table. Also equal to 1/2 the number of the lattice points lying within a prime(n) X prime(n+1) rectangle.

1, 4, 12, 30, 60, 96, 144, 198, 308, 420, 540, 720, 840, 966, 1196, 1508, 1740, 1980, 2310, 2520, 2808, 3198, 3608, 4224, 4800, 5100, 5406, 5724, 6048, 7056, 8190, 8840, 9384, 10212, 11100, 11700, 12636, 13446, 14276, 15308, 16020, 17100, 18240, 18816

Offset: 1

Author

Matthew Howells (mathmatt(AT)gmail.com), Nov 17 2004

Keywords

• nonn

Examples

			a(2) = 4. Since prime(2) is 3 and prime(2+1) is 5, we are playing on a 3x5 billiard table. A ball struck from one corner will cross its own path 4 times before it strikes another corner to return along its own path.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A083553, A087427.

Programs

Mathematica

list = {}; For[i = 1, i < 100, i++, AppendTo[list, (Prime[i] - 1)(Prime[i + 1] - 1)/2]]; list
((First[#]-1)(Last[#]-1))/2&/@Partition[Prime[Range[50]],2,1] (* Harvey P. Dale, Nov 13 2013 *)

Formula

a(n) = (prime(n) - 1)*(prime(n+1) - 1)/2.