A383845 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 elimination pattern is eliminate-eliminate-skip.

1, 1, 2, 1, 2, 3, 1, 2, 4, 3, 1, 2, 4, 5, 3, 1, 2, 4, 5, 3, 6, 1, 2, 4, 5, 7, 3, 6, 1, 2, 4, 5, 7, 8, 6, 3, 1, 2, 4, 5, 7, 8, 3, 6, 9, 1, 2, 4, 5, 7, 8, 10, 3, 9, 6, 1, 2, 4, 5, 7, 8, 10, 11, 6, 9, 3, 1, 2, 4, 5, 7, 8, 10, 11, 3, 6, 12, 9, 1, 2, 4, 5, 7, 8, 10, 11, 13, 3, 9, 12, 6

Offset: 1

Author

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, May 12 2025

Keywords

• nonn
• tabl

Comments

This Josephus problem is related to down-down-under card dealing.

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 pointer eliminates the first number, eliminates the second, then skips the third. The process repeats until no numbers remain. This sequence represents the triangle T(n, k), where n is the number of people in the circle, and T(n, k) is the elimination order of the k-th number in the circle.

Examples

			Consider 4 people in a circle. Initially, person number 1 is eliminated, person number 2 is eliminated, and person number 3 is skipped. The remaining people are now in order 4, 3. Then, both are eliminated. Thus, the fourth row of the triangle is 1, 2, 4, 3, the order of elimination.
The triangle begins
  1;
  1, 2;
  1, 2, 3;
  1, 2, 4, 3;
  1, 2, 4, 5, 3;
  1, 2, 4, 5, 3, 6;
  1, 2, 4, 5, 7, 3, 6;
  1, 2, 4, 5, 7, 8, 6, 3;
		

Crossrefs

Cf. A383846 (row end), A383847 (inverse permutation), A384753 (permutation order).

Cf. A006257, A001651.

Programs

Python

def row(n):
    i, J, out = 0, list(range(1, n+1)), []
    while len(J) > 1:
        i = i%len(J)
        out.append(J.pop(i))
        i = i%len(J)
        if len(J) > 1:
            out.append(J.pop(i))
        i = (i + 1)%len(J)
    out += [J[0]]
    return out
print([e for n in range(1, 14) for e in row(n)])