

A325594


Last remaining numbers after a symmetric variation of the Josephus problem.


1



1, 1, 3, 4, 3, 6, 1, 3, 9, 1, 11, 5, 11, 7, 9, 14, 5, 12, 7, 12, 11, 14, 9, 22, 5, 20, 7, 28, 3, 30, 1, 11, 25, 9, 27, 5, 35, 7, 33, 3, 41, 1, 43, 5, 43, 7, 41, 19, 33, 17, 35, 13, 43, 15, 41, 27, 33, 25, 35, 29, 35, 31
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

A variant of the Josephus problem where two numbers are eliminated at every stage, one elimination clockwise, the other counterclockwise. To resolve ambiguities, the usual Josephus problem takes precedence.


LINKS

Table of n, a(n) for n=1..62.
Hiroshi Matsui, Toshiyuki Yamauchi, Soh Tatsumi, Takahumi Inoue, Masakazu Naito and Ryohei Miyadera, Interesting Variants of the Josephus Problem, Computer Algebra  Design of Algorithms, Implementations and Applications, Kokyuroku, The Research Institute of Mathematical Science, No. 1652, (2009), 4454.
Masakazu Naito and Ryohei Miyadera, The Josephus Problem in Both Directions, The Wolfram Demonstrations Project.
Masakazu Naito, Sohtaro Doro, Daisuke Minematsu and Ryohei Miyadera, The SelfSimilarity of the Josephus Problem and its Variants, Visual Mathematics, Volume 11, No. 2, 2009.
Index entries for sequences related to the Josephus Problem


FORMULA

A165556(n) = a(n) mod 2.


MATHEMATICA

joseboth[m_, mm_] := Block[{t, p, q, u, v, w}, w = mm  1; t = Range[m]; p = t; q = t; Do[p = RotateLeft[p, w]; u = First[p]; p = Rest[p]; q = Drop[q, Position[q, u][[1]]]; If[Length[p] == 1, Break[], ]; q = RotateRight[q, w]; v = Last[q]; q = Drop[q, 1]; p = Drop[p, Position[p, v][[1]]]; If[Length[q] == 1, Break[], ], {n, 1, Ceiling[m/2]}]; p[[1]]];


CROSSREFS

Cf. A006257, A165556.
Sequence in context: A045997 A058024 A262150 * A104076 A238161 A332880
Adjacent sequences: A325591 A325592 A325593 * A325595 A325596 A325597


KEYWORD

nonn


AUTHOR

Gordon Atkinson, Sep 07 2019


STATUS

approved



