A206602 a(n) is the number of white and black stones lying in a circle; starting with place 1, the first a(n) steps of length 5 give the places of white stones. Beginning with last place, the next a(n) steps give the places of black stones.

2, 5, 11, 14, 36, 57, 141, 221, 346, 677, 4042, 9870, 114916, 179557, 1070250, 2612917, 9967491, 12459364, 19467757, 30418371, 38022964, 59410882, 116036880, 283293166, 553306966, 864542135, 1080677669, 3297966522, 8051676081, 15725929847, 19657412309, 47991729272

Offset

1,1

Comments

The game eliminates a(n) - 1 white and a(n) black stones; the a(n)-th white stone survives. The game is known under different names; e.g., "Sankt Peters Spiel" or "Ludus Sancti Petri" or "Josephus problem".

It gives sequences for every step length s from 2.

s=2: 2^n - 1;

s=3: 3, 5, 8, 30, 69, 104, 354, 798, 1797, 2696, 9102, 20481.

s=5: this sequence.

Apparently, this is Seki Takakazu's sequence of "limitative numbers" with m = 5. - Petros Hadjicostas, Jul 18 2020

Links

Table of n, a(n) for n=1..32.

R. Baumann, Computer Knobelei, LOG IN, Heft Nr. 165, pp. 68-71, 2010 (in German).

Saburô Uchiyama, On the generalized Josephus problem, Tsukuba J. Math. 27(2) (2003), 319-339; see p. 337. [Has about 50 sequences related to Seki Takakazu's "limitative numbers"]

Saburô Uchiyama, On the generalized Josephus problem, Tsukuba J. Math. 27(2) (2003), 319-339 [jstor stable version]; see p. 337.

Wikipedia, Seki Takakazu.

Index entries for sequences related to the Josephus Problem

Example

The solution for a(3) = 11: (WBWBWBWBWWBBBWWWBBBWBW);
White stones: (5, 10, 15, 20, 3, 9, 16, 22, 7, 14, 1);
Black stones: (8, 18, 4, 17, 6, 21, 13, 12, 20, 2, 11).

Maple

s:=5: s1:=s-1: a:=1:
for p from 2 to 100000 by 2 do
  b:=(a+s1) mod p +1:
  if (b=1) then printf("%9d", p-1): end if:
  a:=(b+s1) mod (p+1) +1:
if (a=1) then printf("%9d", p): end if:
end do:

Crossrefs

Sequence in context: A331949 A003420 A356426 * A338013 A336190 A238159

Adjacent sequences: A206599 A206600 A206601 * A206603 A206604 A206605

Keyword

nonn

Author

Paul Weisenhorn, Feb 10 2012

Status

approved