Josephus problem?

[Richard Guy]

Why do TWO people count ?      R.

On Wed, 23 Apr 2003, N. J. A. Sloane wrote:

> I looked up the original definition of A005428,
> and revised the entry:
> 
> 
> %I A005428 M0572
> %S A005428 1,2,3,4,6,9,14,21,31,47,70,105,158,237,355,533,799,1199,1798,2697,4046,6069,9103,
> %T A005428 13655,20482,30723,46085,69127,103691,155536,233304,349956,524934,787401
> %N A005428 a(0) = 1, state(0) = 1; for n >= 1, if a(n-1) is even then a(n) = floor( 3*a(n-1)/2 ) and state(n) = state(n-1), but if a(n-1) is odd then a(n) = ceiling( 3*a(n-1)/2 ) and state(n) = 3 - state(n-1).
> %C A005428 Arises from a version of the Josephus problem: two people, count 1, 2, 3, and every pers
> on indicated by 3 drops out.
> %D A005428 K. Burde, Das Problem der Abzahlreime und Zahlentwicklungen mit gebrochenen Basen, J. Number Theory 26 (1987), no. 2, 192-209.
> %D A005428 F. Schuh, The Master Book of Mathematical Recreations. Dover, NY, 1968, page, 374, Table
>  18, union of coulumns 1 and 2.
> %Y A005428 Cf. A005427. Union of A081614 and A081615.
> %Y A005428 Is this the same as log2(A082125(n+3))?. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 16 2002
> %Y A005428 Is this the same as A073941? - Benoit Cloitre, Nov 24, 2002
> %K A005428 nonn,more,easy
> %O A005428 0,2
> %A A005428 njas and Simon Plouffe (plouffe(AT)math.uqam.ca)
> %E A005428 Entry revised Apr 23, 2002.
> 
> %I A081614
> %S A081614 1,4,6,9,31,70,105
> %N A081614 Subsequence of A005428 with state = 1.
> %O A081614 0,2
> %K A081614 nonn,easy,more
> %Y A081614 Cf. A005428, A081615.
> %A A081614 njas, Apr 23 2003
> 
> %I A081615
> %S A081615 2,3,14,21,47,158,237
> %N A081615 Subsequence of A005428 with state = 2.
> %O A081615 0,2
> %K A081615 nonn,easy,more
> %Y A081615 Cf. A005428, A081614.
> %A A081615 njas, Apr 23 2003
> 
> NJAS
>