A165735 Surviving integers under the double-count Josephus problem (see A054995), modulo 3.

1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,2

Comments

Old name was: The pattern is obvious. The sequence can be divided into subsequences of {1,1,1,...} and {2,2,2,...}.

Let n be a natural number. We put n numbers in a circle, and we are going to remove every third number. Let J3(n) be the last number that remains. This is the traditional Josephus Problem. Let J3 (mod 3) be the residue of the sequence J3(n) under mod 3. J3 (mod 3) produces the sequence {1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2,...}.

Links

Table of n, a(n) for n=1..105.

Hiroshi Matsui, Masakazu Naito and Naoyuki Totani, The Period and the Distribution of the Fibonacci-like Sequence Under Various Moduli, Undergraduate Math Journal, Rose-Hulman Institute of Technology, Vol. 10, Issue 1, 2009.

Masakazu Naito and Ryohei Miyadera, The Self-Similarity of the Josephus Problem and its Variants, Visual Mathematics, Volume 11, No.2, 2009.

Wolfram MathWorld, Josephus Problem

Index entries for sequences related to the Josephus Problem

Formula

(1) J3(1) = 1 and J3(2) = 2.

(2) J3(3m) = J3(2m) + [(J3(2m)-1)/2].

(3a) J3(3m+1) = 3m + 1 (if J3(2m + 1) = 1).

(3b) J3(3m+1) = J3(2m+1) + [J3(2m+1)/2] - 2 (if J3(2m + 1) > 1).

(4) J3(3m+2) = J3(2m+1) + [J3(2m+1)/2] + 1

a(n) = A010872(A054995(n)). - Gordon Atkinson, Aug 21 2019

Example

If we use n = 10, then we put numbers 1,2,3,4,5,6,7,8,9,10 in a circle. We eliminate 3,6,9,2,7,1,8,5,10, and the last number that remains is 4. Therefore J3(10) = 4 and J3(10) = 1 mod 3.

Mathematica

J3[1] = 1; J3[2] = 2; J3[n_] := J3[n] = Block[{m, t}, t = Mod[n, 3]; m = (n - t)/3; Which[t == 0, J3[2 m] + Floor[(J3[2 m] - 1)/2], t == 1, If[J3[2 m + 1] == 1, 3 m + 1, J3[2 m + 1] + Floor[J3[2 m + 1]/2] - 2], t == 2, J3[2 m + 1] + Floor[J3[2 m + 1]/2] + 1]]; Table[Mod[J3[n], 3], {n, 1, 200}]

Crossrefs

Cf. A010872, A054995, A114144, A113648, A165556.

Sequence in context: A098357 A370958 A335208 * A083888 A102681 A274013

Adjacent sequences: A165732 A165733 A165734 * A165736 A165737 A165738

Keyword

nonn

Author

Ryohei Miyadera and Masakazu Naito, Sep 25 2009

Extensions

New name from Gordon Atkinson, Aug 21 2019

Status

approved