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.

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

1,2

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(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.

Links

William Rex Marshall, First 141 antidiagonals of array, flattened

Index entries for sequences related to the Josephus Problem

Formula

T(k,1) = 1;

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

Example

.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

Crossrefs

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.

Sequence in context: A122610 A011973 A115139 * A112543 A099478 A133913

Adjacent sequences: A198785 A198786 A198787 * A198789 A198790 A198791

Keyword

nonn,easy,tabl

Author

William Rex Marshall, Nov 21 2011

Status

approved