A341216 Triangle read by columns T(n,k) k > n >= 1: Last survivor positions in a modified Josephus problem for n numbers, where after each deletion the counting starts over at the lowest existing number n, rather than continuing from the current position.

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

Offset: 1

Author

Gary Yane, Feb 06 2021

Keywords

• nonn
• tabl

Comments

Arrange 1,2,3,...,n clockwise in a circle. Start the count at the lowest surviving value and delete the k-th value counting clockwise around the circle. Repeat this procedure until one number remains, which is T(n,k).

Note: In the complete n X k array with n >= 1 and k >= 1, T(n,k) = T(k-1,k) for all n >= k > 1 and T(n,1)=n.

That makes the bottom triangle of the array unchanging, so it is omitted.

Examples

			n\k    2    3    4    5    6    7    8    9   10   11   12   13
_______________________________________________________________
1      1    1    1    1    1    1    1    1    1    1    1    1
2           2    1    2    1    2    1    2    1    2    1    2
3                2    3    1    3    1    2    2    3    1    3
5                     4    1    4    1    3    3    4    1    4
6                          2    5    1    3    3    5    1    5
7                               6    1    4    3    6    1    6
8                                    2    5    4    7    1    7
9                                         6    5    8    1    8
10                                             6    9    1    9
11                                                 10    1   10
12                                                       2   11
13                                                           12
		

Links

• Index entries for sequences related to the Josephus Problem

Crossrefs

The last entry in each column is A128982.

Formula

T(1,k) = 1, for k > 1;

T(n,k) = T(n-1,k) if k mod n > T(n-1,k) or k mod n = 0;

T(n,k) = T(n-1,k) + 1 otherwise.