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A381591 Elimination order of the first person in a variation of the Josephus problem, where there are n people total and two people are skipped each time.

%I A381591 #20 Mar 26 2025 22:02:08
%S A381591 1,1,2,4,2,6,6,3,9,6,4,7,11,5,11,15,6,13,11,7,12,16,8,23,18,9,22,16,
%T A381591 10,17,31,11,27,30,12,35,21,13,22,37,14,30,35,15,32,26,16,27,35,17,47,
%U A381591 37,18,53,31,19,32,47,20,57,56,21,51,36,22,37,65,23,49,70
%N A381591 Elimination order of the first person in a variation of the Josephus problem, where there are n people total and two people are skipped each time.
%C A381591 a(3k-1) = k.
%e A381591 Consider n = 4 people. The first person eliminated is number 3. This leaves the remaining people in order 4, 1, 2. The second person eliminated is number 2. Thus, the remaining people in order 4, 1. The next person eliminated is number 4. On the fourth step, person number 1 is eliminated, implying that the order of elimination of the first person is 4: a(4) = 4.
%o A381591 (Python)
%o A381591 def UUD(n):
%o A381591     return invPerm(UUDES(n))
%o A381591 def UUDES(n):
%o A381591     l=[]
%o A381591     for i in range(n):
%o A381591         l.append(i+1)
%o A381591     index = 0
%o A381591     P=[]
%o A381591     for i in range(n):
%o A381591         index+=2
%o A381591         index=index%len(l)
%o A381591         P.append(l[index])
%o A381591         l.pop(index)
%o A381591     return P
%o A381591 def invPerm(p):
%o A381591     inv = []
%o A381591     for i in range(len(p)):
%o A381591         inv.append(None)
%o A381591     for i in range(len(p)):
%o A381591         inv[p[i]-1]=i+1
%o A381591     return inv
%o A381591 sequence = []
%o A381591 for i in range(1, 71):
%o A381591     sequence += [str(UUD(i)[0])]
%o A381591 print(", ".join(sequence))
%o A381591 (Python)
%o A381591 def a(n):
%o A381591     c, i, J = 1, 0, list(range(1, n+1))
%o A381591     while len(J) > 0:
%o A381591         i = (i + 2)%len(J)
%o A381591         q = J.pop(i)
%o A381591         if q == 1: return c
%o A381591         c = c+1
%o A381591 print([a(n) for n in range(1, 71)]) # _Michael S. Branicky_, Mar 24 2025
%Y A381591 Cf. A006257, A054995, A225381, A321298, A378635, A380195, A008585, A381591 A381667.
%K A381591 nonn
%O A381591 1,3
%A A381591 _Tanya Khovanova_, _Nathan Sheffield_, and the MIT PRIMES STEP junior group, Mar 02 2025