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A317626 Intersections with the x-axis of a bouncing ball on a Sophie Germain billiard table. 0
2, 4, 8, 10, 14, 18, 28, 30, 38, 44, 58, 60, 64, 78, 80, 84, 94, 98, 120, 140, 144, 148, 164, 170, 198, 214, 218, 220, 228, 240, 248, 254, 270, 304, 318, 338, 340, 344, 350, 368, 408, 410, 430, 470, 480, 484, 494, 500, 504, 520, 528, 534, 578, 604, 630, 634, 644, 658 (list; graph; refs; listen; history; text; internal format)
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.

The sum of a term of this sequence and the corresponding Sophie Germain prime is A317510 and it appears that this is a subsequence of A179882. Checked up to and including 33295 of A317510 (Sophie Germain prime 24971).

LINKS

Table of n, a(n) for n=1..58.

Samuel King, Billiard Simulator

Hilko Koning, Some bouncing examples.

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

Cf. A005384, A151922, A176045, A124485, A179882, A317510.

Sequence in context: A050567 A069879 A074330 * A292550 A024895 A250310

Adjacent sequences:  A317623 A317624 A317625 * A317627 A317628 A317629

KEYWORD

nonn

AUTHOR

Hilko Koning, Aug 02 2018

STATUS

approved

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Last modified September 17 03:42 EDT 2021. Contains 347478 sequences. (Running on oeis4.)