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

1,3

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.

Links

Table of n, a(n) for n=1..105.

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

Example

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.

Sequence in context: A085014 A082074 A132283 * A378674 A307081 A256440

Adjacent sequences: A378679 A378680 A378681 * A378685 A378686 A378687

Keyword

nonn,tabl,new

Author

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

Status

approved