A114144 A variant of the Josephus Problem in which three persons are to be eliminated at the same time.

3, 1, 3, 5, 8, 11, 14, 17, 21, 25, 29, 33, 37, 41, 45, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 133

Offset

1,1

Comments

This is a variant of the Josephus Problem. When there are 3m persons, the first process of elimination starts with the first person, the second with the (m+1)-st person and the third with the (2m+1)-st person. We suppose that the first process comes first, the second process secondly and the third process thirdly. J(n) is the position of the survivor when there are n persons. Our sequence is a(n) = J(3*n).

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley Publishing Company, 1994, pp. 9-10.

Links

Table of n, a(n) for n=1..65.

Index entries for sequences related to the Josephus Problem

Formula

The function J(n) is defined only for integers n that have 3 as a factor. J(6m+3) = 2J(3m)+2m+2 (if J(3m) <= m), J(6m+3) = 2J(3m)+2m+3 (if m+1 <= J(3m) <= 2m) and J(6m+3) = 2J(3m)-4m+1 (if 2m+1 <= J(3m)). J(6m) = 2J(3m)+2m-1 (if J(3m) <= 2m) and J(6m) = 2J(3m)-4m-1 (if J(3m) > 2m).

Example

If there are 15 persons, then 2, 7, 12, 4, 9, 14, 6, 11, 1, 10, 15, 5, 3, 13 are to be eliminated and the survivor is 8. Therefore a(5) = J(15) = 8.

Mathematica

(*This function is defined only for numbers that are multiples of 3.*)
jose[3] = 3; jose[n_?(IntegerQ[ #/3] &)] := If[Mod[n, 6] == 0, If[jose[n/2] < n/3 + 1, 2jose[n/2] + n/3 - 1, 2jose[n/2] - 2n/3 - 1], Which[jose[(n - 3)/2] < (n - 3)/6 +1, 2jose[(n - 3)/2] + (n - 3)/3 + 2, (n - 3)/6 < jose[(n - 3)/2] < (n - 3)/3 + 1, 2jose[(n - 3)/2] + (n - 3)/3 + 3, (n - 3)/3 < jose[(n - 3)/2], 2jose[(n - 3)/2] - 2(n - 3)/3 + 1]];

Prog

(PARI) a(n) = if(n==0, 1, my(t); if(n%2, t=a(n\2); if(t>n-1, 2*t-2*n+3, if(t>n\2, 2*t+n+2, 2*t+n+1)), t=a(n/2); if(t>n, 2*t-2*n-1, 2*t+n-1))); \\ Jinyuan Wang, Apr 20 2025

Crossrefs

Cf. A006257, A113648.

Sequence in context: A234587 A339413 A388919 * A050820 A261869 A279697

Adjacent sequences: A114141 A114142 A114143 * A114145 A114146 A114147

Keyword

nonn,easy

Author

Satoshi Hashiba, Daisuke Minematsu and Ryohei Miyadera, Feb 03 2006

Extensions

More terms from Jinyuan Wang, Apr 20 2025

Status

approved