A178853 -id:A178853 - cp's OEIS frontend

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

William Rex Marshall, Nov 21 2011

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.

			.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

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

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.

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 *)

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

A198788 Array T(k,n) read by descending antidiagonals: Last survivor positions in Josephus problem for n numbers and a count of k, n >= 1, k >= 1.

Original entry on oeis.org

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

Offset: 1

William Rex Marshall, Nov 21 2011

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(k,n).
The main diagonal of the array (1, 1, 2, 2, 2, 4, 5, 4, ...) is A007495.
Consecutive columns down to the main diagonal (1, 2, 1, 3, 3, 2, 4, 1, 1, 2, ...) is A032434.
Period lengths of columns (1, 2, 6, 12, 60, 60, 420, 840, ...) is A003418.

			.k\n  1  2  3  4  5  6  7  8  9 10
----------------------------------
.1 |  1  2  3  4  5  6  7  8  9 10  A000027
.2 |  1  1  3  1  3  5  7  1  3  5  A006257
.3 |  1  2  2  1  4  1  4  7  1  4  A054995
.4 |  1  1  2  2  1  5  2  6  1  5  A088333
.5 |  1  2  1  2  2  1  6  3  8  3  A181281
.6 |  1  1  1  3  4  4  3  1  7  3
.7 |  1  2  3  2  4  5  5  4  2  9  A178853
.8 |  1  1  3  3  1  3  4  4  3  1  A109630
.9 |  1  2  2  3  2  5  7  8  8  7
10 |  1  1  2  4  4  2  5  7  8  8

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

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

Cf. A003418, A007495 (main diagonal), A032434, A198789, A198790.

T(k,1) = 1;

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

cp's OEIS Frontend

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.

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