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A378682 Triangle T(n,k) read by rows: where T(n,k) is the number of the k-th eliminated person in the variation of the Josephus elimination process for n people, where the first person is eliminated, and after that, every second person is eliminated.

%I A378682 #10 Dec 12 2024 15:22:00
%S A378682 1,1,2,1,3,2,1,3,2,4,1,3,5,4,2,1,3,5,2,6,4,1,3,5,7,4,2,6,1,3,5,7,2,6,
%T A378682 4,8,1,3,5,7,9,4,8,6,2,1,3,5,7,9,2,6,10,8,4,1,3,5,7,9,11,4,8,2,10,6,1,
%U A378682 3,5,7,9,11,2,6,10,4,12,8,1,3,5,7,9,11,13,4,8,12,6,2,10,1,3,5,7,9,11,13,2,6,10,14,8,4,12
%N A378682 Triangle T(n,k) read by rows: where T(n,k) is the number of the k-th eliminated person in the variation of the Josephus elimination process for n people, where the first person is eliminated, and after that, every second person is eliminated.
%C A378682 The n-th row has n elements.
%C A378682 In this variation of the Josephus elimination process, the numbers 1 through n are arranged in a circle. A pointer starts at position 1. With each turn, the number at the pointer is eliminated, and the pointer skips the next number. This repeats until no numbers remain. This sequence represents the triangle J(n, k), where n is the number of people in the circle, and J(n, k) is the elimination order of the k-th number in the circle.
%F A378682 T(n,1) = 1 and T(n,k) = A321298(n-1,k-1) + 1, for n,k > 1.
%F A378682 T(n,k) = (A321298(n,k)-2 mod n) + 1. - _Pontus von Brömssen_, Dec 11 2024
%e A378682 Consider 5 people in a circle. During the first round around the circle, people numbered 1, 3, and 5 are eliminated in this order. The next person, numbered 2, is skipped, and 4 is eliminated. Person 2 is eliminated last. Thus, the fifth row of the triangle is 1, 3, 5, 4, 2.
%e A378682 Triangle begins;
%e A378682   1;
%e A378682   1, 2;
%e A378682   1, 3, 2;
%e A378682   1, 3, 2, 4;
%e A378682   1, 3, 5, 4, 2;
%e A378682   1, 3, 5, 2, 6, 4;
%e A378682   1, 3, 5, 7, 4, 2, 6;
%e A378682   1, 3, 5, 7, 2, 6, 4, 8;
%e A378682   1, 3, 5, 7, 9, 4, 8, 6, 2;
%Y A378682 Cf. A006257, A225381, A321298, A378635, A378674.
%K A378682 nonn,tabl
%O A378682 1,3
%A A378682 _Tanya Khovanova_ and the MIT PRIMES STEP junior group, Dec 03 2024