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A384772 Triangle read by row: T(n,k) is the number of the k-th eliminated person in a variant of the Josephus problem in which first three people are skipped, then one is eliminated, repeating until all n people are eliminated.

%I A384772 #14 Jul 23 2025 16:09:51
%S A384772 1,2,1,1,3,2,4,1,3,2,4,3,5,2,1,4,2,1,3,6,5,4,1,6,5,7,3,2,4,8,5,2,1,3,
%T A384772 7,6,4,8,3,9,6,5,7,2,1,4,8,2,7,3,10,9,1,6,5,4,8,1,6,11,7,3,2,5,10,9,4,
%U A384772 8,12,5,10,3,11,7,6,9,2,1,4,8,12,3,9,1,7,2,11,10,13,6,5
%N A384772 Triangle read by row: T(n,k) is the number of the k-th eliminated person in a variant of the Josephus problem in which first three people are skipped, then one is eliminated, repeating until all n people are eliminated.
%C A384772 The numbers 1 through n are arranged in a circle. A pointer starts at position 1. With each turn, the pointer skips three numbers and the next number is eliminated. The process repeats until no numbers remain. This sequence is a concatenation of the rows of the triangle T, where T(n, k) is the elimination order of the k-th number in a circle of n  numbers.
%C A384772 This variation of the Josephus problem can equivalently be described in terms of under-under-under-down card dealing, where the cards of a deck are dealt by alternately cycling three cards from the top "under", and then dealing the next card "down". In particular, T(n,k) is the k-th card dealt in under-under-under-down dealing if the deck begins in order 1,2,3,...,n.
%e A384772 Consider 4 people in a circle. Initially, people numbered 1, 2, and 3 are skipped, and person 4 is eliminated. The remaining people are now in order 1, 2, 3. Then all three are skipped and person 1 is eliminated. The remaining people are in order 2, 3. Now, we skip over 2, 3, 2 and eliminate person 3. Person 2 is eliminated last. Thus, the fourth row of the triangle is 4, 1, 3, 2.
%e A384772 The triangle begins as follows:
%e A384772   1;
%e A384772   2, 1;
%e A384772   1, 3, 2;
%e A384772   4, 1, 3, 2;
%e A384772   4, 3, 5, 2, 1;
%e A384772   4, 2, 1, 3, 6, 5;
%e A384772   4, 1, 6, 5, 7, 3, 2;
%e A384772   4, 8, 5, 2, 1, 3, 7, 6;
%e A384772   4, 8, 3, 9, 6, 5, 7, 2, 1
%o A384772 (Python)
%o A384772 def row(n):
%o A384772     i, J, out = 0, list(range(1, n+1)), []
%o A384772     while len(J) > 0:
%o A384772         i = (i + 3)%len(J)
%o A384772         out.append(J.pop(i))
%o A384772     return out
%o A384772 print([e for n in range(1, 14) for e in row(n)])
%Y A384772 Cf. A378635, A321298, A006257, A225381, A008586, A384770, A088333, A384774.
%K A384772 nonn,tabl
%O A384772 1,2
%A A384772 _Tanya Khovanova_, _Nathan Sheffield_, and the MIT PRIMES STEP junior group, Jun 09 2025