A198789 Array T(n,k) read by antidiagonals: Last survivor positions in Josephus problem for n numbers and a count of k, n >= 1, k >= 1.

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

Offset: 1

Author

William Rex Marshall, Nov 21 2011

Keywords

• nonn
• easy
• tabl

Comments

Arrange 1, 2, 3, ..., n clockwise in a circle. Starting the count at 1, delete every k-th integer clockwise until only one remains, which is T(n,k).

The main diagonal (1, 1, 2, 2, 2, 4, 5, 4, ...) is A007495.

Concatenation of consecutive rows (up to the main diagonal) gives A032434.

The periods of the rows, (1, 2, 6, 12, 60, 60, 420, 840, ...), is given by A003418.

Examples

			.n\k  1  2  3  4  5  6  7  8  9 10
----------------------------------
.1 |  1  1  1  1  1  1  1  1  1  1
.2 |  2  1  2  1  2  1  2  1  2  1
.3 |  3  3  2  2  1  1  3  3  2  2
.4 |  4  1  1  2  2  3  2  3  3  4
.5 |  5  3  4  1  2  4  4  1  2  4
.6 |  6  5  1  5  1  4  5  3  5  2
.7 |  7  7  4  2  6  3  5  4  7  5
.8 |  8  1  7  6  3  1  4  4  8  7
.9 |  9  3  1  1  8  7  2  3  8  8
10 | 10  5  4  5  3  3  9  1  7  8
		

Links

• William Rex Marshall, First 141 antidiagonals of array, flattened
• Index entries for sequences related to the Josephus Problem

Crossrefs

Cf. A000027 (k = 1), A006257 (k = 2), A054995 (k = 3), A088333 (k = 4), A181281 (k = 5), A360268 (k = 6), A178853 (k = 7), A109630 (k = 8).

Cf. A003418, A007495 (main diagonal), A032434, A198788, A198790.

Programs

Mathematica

T[n_, k_] := T[n, k] = If[n == 1, 1, Mod[T[n-1, k]+k-1, n]+1];
Table[T[n-k+1, k], {n, 1, 13}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 04 2023 *)

Formula

T(1,k) = 1; for n > 1: T(n,k) = ((T(n-1,k) + k - 1) mod n) + 1.