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A032436
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Triangle of third-to-last man to survive in the Josephus problem of n men in a circle with every k-th killed, with 1 <= k <= n and n >= 3.
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3
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1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 4, 1, 1, 3, 1, 2, 5, 3, 1, 2, 1, 1, 2, 6, 1, 4, 3, 3, 1, 1, 2, 7, 3, 1, 1, 2, 4, 1, 1, 2, 8, 1, 4, 1, 3, 3, 5, 1, 1, 4, 9, 3, 2, 5, 1, 5, 1, 1, 4, 3, 2, 10, 1, 5, 1, 1, 3, 8, 2, 1, 1, 1, 2, 11, 3, 1, 5, 6, 4, 2, 4, 3, 1, 1, 1, 7, 12, 5, 2, 3, 2, 1, 9, 4, 5, 7, 1, 1, 6
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OFFSET
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3,4
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REFERENCES
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W. W. R. Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover, pp. 32-36, 1987.
M. Kraitchik, "Josephus' Problem," Sec. 3.13 in Mathematical Recreations, New York: W. W. Norton, pp. 93-94, 1942.
Eric W. Weisstein, The CRC Concise Encyclopedia in Mathematics, 2nd ed., Chapman and Hall/CRC, 2002. [The first 7 rows of the triangle appear on p. 1596 of this book under the topic "Josephus Problem".]
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LINKS
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F. Jakóbczyk, On the generalized Josephus problem, Glasow Math. J. 14(2) (1973), 168-173. [It contains algorithms that allow the identification of the original position of the third-to-last person to survive in Josephus problem.]
Eric Weisstein's World of Mathematics, Josephus Problem. [It contains a new, apparently corrected, triangle.]
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EXAMPLE
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Triangle T(n,k) (with rows n >= 3 and columns k = 1..n) begins
1, 1, 1;
2, 1, 1, 1;
3, 1, 2, 1, 2;
4, 1, 1, 3, 1, 2;
5, 3, 1, 2, 1, 1, 2;
6, 1, 4, 3, 3, 1, 1, 2;
7, 3, 1, 1, 2, 4, 1, 1, 2;
8, 1, 4, 1, 3, 3, 5, 1, 1, 4;
9, 3, 2, 5, 1, 5, 1, 1, 4, 3, 2;
10, 1, 5, 1, 1, 3, 8, 2, 1, 1, 1, 2;
11, 3, 1, 5, 6, 4, 2, 4, 3, 1, 1, 1, 7;
...
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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