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

%I A114144 #23 Apr 21 2025 02:42:09
%S A114144 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,
%T A114144 23,25,27,29,31,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84,
%U A114144 87,90,93,96,99,102,105,108,111,114,117,120,123,126,129,133
%N A114144 A variant of the Josephus Problem in which three persons are to be eliminated at the same time.
%C A114144 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).
%D A114144 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley Publishing Company, 1994, pp. 9-10.
%H A114144 <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%F A114144 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).
%e A114144 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.
%t A114144 (*This function is defined only for numbers that are multiples of 3.*)
%t A114144 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]];
%o A114144 (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
%Y A114144 Cf. A006257, A113648.
%K A114144 nonn,easy
%O A114144 1,1
%A A114144 Satoshi Hashiba, Daisuke Minematsu and _Ryohei Miyadera_, Feb 03 2006
%E A114144 More terms from _Jinyuan Wang_, Apr 20 2025