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.

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, 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, 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

Offset: 1

Author

Tanya Khovanova and the MIT PRIMES STEP junior group, Dec 03 2024

Keywords

• nonn
• tabl

Comments

The n-th row has n elements.

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.

Examples

			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.
Triangle begins;
  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, 4, 8;
  1, 3, 5, 7, 9, 4, 8, 6, 2;
		

Crossrefs

Cf. A006257, A225381, A321298, A378635, A378674.

Formula

T(n,1) = 1 and T(n,k) = A321298(n-1,k-1) + 1, for n,k > 1.

T(n,k) = (A321298(n,k)-2 mod n) + 1. - Pontus von Brömssen, Dec 11 2024