A386641 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 one person is skipped, then one is eliminated, then two people are skipped, then the next person is eliminated and so on.

1, 2, 1, 2, 3, 1, 2, 1, 4, 3, 2, 5, 1, 3, 4, 2, 5, 4, 1, 6, 3, 2, 5, 3, 4, 1, 6, 7, 2, 5, 1, 8, 4, 6, 3, 7, 2, 5, 9, 7, 8, 4, 1, 3, 6, 2, 5, 9, 6, 4, 8, 7, 3, 1, 10, 2, 5, 9, 4, 1, 3, 8, 10, 11, 6, 7, 2, 5, 9, 3, 11, 10, 1, 8, 4, 7, 6, 12, 2, 5, 9, 1, 10, 7, 8, 13, 12, 6, 4, 11, 3, 2, 5, 9, 14, 8, 4, 3, 7, 13

Offset

1,2

Comments

The numbers 1 through n are arranged in a circle. The process starts at position 1. Initially, the first number is skipped, and the next number is eliminated. Then, two numbers are skipped, and the next one is eliminated. Then, three numbers are skipped, and so on. The process repeats until no numbers remain.

This variation of the Josephus problem can equivalently be described in terms of the AP card dealing, where the cards of a deck are dealt by alternately x cards from the top "under", and then dealing the next card "down". Here, x starts as 1, and is increased by 1 with every dealt card. In particular, T(n,k) is the k-th card dealt in the AP dealing if the deck begins in order 1,2,3,...,n.

The freed person is A291317(n).

Links

Table of n, a(n) for n=1..100.

Eric Huang, Tanya Khovanova, Timur Kilybayev, Ryan Li, Brandon Ni, Leone Seidel, Samarth Sharma, Nathan Sheffield, Vivek Varanasi, Alice Yin, Boya Yun, and William Zelevinsky, Card Dealing Math, arXiv:2509.11395 [math.NT], 2025. See pp. 17, 39.

Formula

T(n,k) = A000096(k), when n >= A000096(k).

Example

Consider 4 people in a circle. Initially, person number 1 is skipped, and person 2 is eliminated. The remaining people are now in order 3, 4, 1. Then, two people are skipped, and person 1 is eliminated. The remaining people are in order 3, 4. Now, three people are skipped and person 4 is eliminated. Person 3 is eliminated last. Thus, the fourth row of the triangle is 2,1,4,3.
The triangle begins as follows:
  1;
  2, 1;
  2, 3, 1;
  2, 1, 4, 3;
  2, 5, 1, 3, 4;
  2, 5, 4, 1, 6, 3;
  2, 5, 3, 4, 1, 6, 7;
  2, 5, 1, 8, 4, 6, 3, 7;
  2, 5, 9, 7, 8, 4, 1, 3, 6;

Prog

(Python)
def row(n):
  c, i, J = 1, 0, list(range(1, n+1))
  output = []
  while len(J) > 1:
    i = (i + c) % len(J)
    q = J.pop(i)
    output.append(q)
    c = c + 1
  output.append(J[0])
  return output
print([e for n in range(1, 15) for e in row(n)])

Crossrefs

Cf. A000096, A291317, A386639, A386643.

Cf. A321298 (classical elimination process).

Sequence in context: A280052 A183198 A249160 * A387618 A346913 A269970

Adjacent sequences: A386638 A386639 A386640 * A386642 A386643 A386644

Keyword

nonn,tabl

Author

Tanya Khovanova, Nathan Sheffield, and the MIT PRIMES STEP junior group, Jul 27 2025

Status

approved