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A198789
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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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6
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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T(1,k) = 1; for n > 1: T(n,k) = ((T(n-1,k) + k - 1) mod n) + 1.
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EXAMPLE
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.n\k 1 2 3 4 5 6 7 8 9 10
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.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
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MATHEMATICA
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T[n_, k_] := T[n, k] = If[n == 1, 1, Mod[T[n-1, k]+k-1, n]+1];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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