OFFSET
1,1
COMMENTS
In the first quadrant of a coordinate system define a rectangular Sophie Germain billiard table with width p and length 2p+1, with vertices (0,0), (p,0), (p,2p+1) and (0,2p+1). A billiard ball (considered to be a point) starts from (0,0) at an angle of 45 degrees and hits the sides exactly p times until it hits the x-axis. The sequence gives the intersections with the x-axis of consecutive Sophie Germain prime numbers (p > 3) after p bounces.
The sum of all crossed lattice points (including the rectangle sides) is the sum of crossed points left under, right middle and left up respectively ((p+7)/6)^2 + (p+1)(p+4)/18 + (p+1)(p+7)/36 = ((p+4)/3)^2 (see bouncing examples).
The enclosed areas in the Sophie Germain billiard table also correspond to ((p+4)/3)^2.
The number of trajectories is a subsequence of A176045.
The number of trajectories with slope +1 or with slope -1 is a subsequence of A124485.
LINKS
FORMULA
a(n) = (A005384(n)+1)/3 for n>=3. - Michel Marcus, Aug 25 2018
MATHEMATICA
lst = {}; Do[If[PrimeQ[p] && PrimeQ[2 p + 1], AppendTo[lst, (p + 1)/3]], {p, 5, 2*10^3}]; lst
(Select[Prime@ Range[3, 300], PrimeQ[2# + 1] &] + 1)/3 (* Robert G. Wilson v, Aug 02 2018 *)
PROG
(PARI) lista(nn) = forprime(p=2, nn, if (isprime(2*p+1), print1((p+1)/3, ", ")); ); \\ Michel Marcus, Aug 25 2018
(GAP) a:=[];; for p in [3..2000] do if IsPrime(p) and IsPrime(2*p+1) then Add(a, (p+1)/3); fi; od; a; # Muniru A Asiru, Aug 28 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Hilko Koning, Aug 02 2018
STATUS
approved