A032436 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.

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

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• tabf

Examples

			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;
  ...
		

References

• 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".]

Links

• W. W. R. Ball, Mathematical Recreations and Essays, 4th ed., New York: The MacMillan Company, 1905 (see "Decimation" on pp. 19-20).
• Sean A. Irvine, A032435 and A032436 Josephus problem data mismatch, message in seqfan, June 2020.
• 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.]
• M. Kraitchik, "Josephus' Problem", Sec. 3.13 in Mathematical Recreations, New York: W. W. Norton, pp. 93-94, 1942. [Available only in the USA through the Hathi Trust Digital Library.]
• Eric Weisstein's World of Mathematics, Josephus Problem. [It contains a new, apparently corrected, triangle.]
• Wikipedia, Josephus problem.
• Index entries for sequences related to the Josephus Problem

Crossrefs

Cf. A032434, A032435.