A198789 - OEIS

%I #32 Mar 27 2023 05:16:54

%S 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,

%T 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,

%U 2,2,3,1,5,3,3,1,4,7,12,1,1,1,4,2,3,5,1,8,5,7,9,13

%N 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.

%C 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).

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

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

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

%H William Rex Marshall, <a href="/A198789/b198789.txt">First 141 antidiagonals of array, flattened</a>

%H <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>

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

%e .n\k  1  2  3  4  5  6  7  8  9 10

%e ----------------------------------

%e .1 |  1  1  1  1  1  1  1  1  1  1

%e .2 |  2  1  2  1  2  1  2  1  2  1

%e .3 |  3  3  2  2  1  1  3  3  2  2

%e .4 |  4  1  1  2  2  3  2  3  3  4

%e .5 |  5  3  4  1  2  4  4  1  2  4

%e .6 |  6  5  1  5  1  4  5  3  5  2

%e .7 |  7  7  4  2  6  3  5  4  7  5

%e .8 |  8  1  7  6  3  1  4  4  8  7

%e .9 |  9  3  1  1  8  7  2  3  8  8

%e 10 | 10  5  4  5  3  3  9  1  7  8

%t T[n_, k_] := T[n, k] = If[n == 1, 1, Mod[T[n-1, k]+k-1, n]+1];

%t Table[T[n-k+1, k], {n, 1, 13}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Mar 04 2023 *)

%Y 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).

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

%K nonn,easy,tabl

%O 1,3

%A _William Rex Marshall_, Nov 21 2011