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A198788
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Array T(k,n) 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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4
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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
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OFFSET
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1,2
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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(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.
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LINKS
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William Rex Marshall, First 141 antidiagonals of array, flattened
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FORMULA
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T(k,1) = 1;
for n > 1: T(k,n) = ((T(k,n-1) + k - 1) mod n) + 1.
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EXAMPLE
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.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
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CROSSREFS
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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
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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William Rex Marshall, Nov 21 2011
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STATUS
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approved
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