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%I #31 Feb 09 2023 14:51:08
%S 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,
%T 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,
%U 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
%N 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.
%D W. W. R. Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover, pp. 32-36, 1987.
%D M. Kraitchik, "Josephus' Problem," Sec. 3.13 in Mathematical Recreations, New York: W. W. Norton, pp. 93-94, 1942.
%D 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".]
%H W. W. R. Ball, <a href="http://www.gutenberg.org/files/26839/26839-pdf.pdf">Mathematical Recreations and Essays</a>, 4th ed., New York: The MacMillan Company, 1905 (see "Decimation" on pp. 19-20).
%H Sean A. Irvine, <a href="http://list.seqfan.eu/oldermail/seqfan/2020-June/020790.html">A032435 and A032436 Josephus problem data mismatch</a>, message in seqfan, June 2020.
%H F. Jakóbczyk, <a href="https://doi.org/10.1017/S0017089500001919">On the generalized Josephus problem</a>, 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.]
%H M. Kraitchik, <a href="https://babel.hathitrust.org/cgi/pt?id=wu.89041209552&view=1up&seq=95">"Josephus' Problem"</a>, Sec. 3.13 in Mathematical Recreations, New York: W. W. Norton, pp. 93-94, 1942. [Available only in the USA through the <a href="https://www.hathitrust.org/">Hathi Trust Digital Library</a>.]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JosephusProblem.html">Josephus Problem</a>. [It contains a new, apparently corrected, triangle.]
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Josephus_problem">Josephus problem</a>.
%H <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%e Triangle T(n,k) (with rows n >= 3 and columns k = 1..n) begins
%e 1, 1, 1;
%e 2, 1, 1, 1;
%e 3, 1, 2, 1, 2;
%e 4, 1, 1, 3, 1, 2;
%e 5, 3, 1, 2, 1, 1, 2;
%e 6, 1, 4, 3, 3, 1, 1, 2;
%e 7, 3, 1, 1, 2, 4, 1, 1, 2;
%e 8, 1, 4, 1, 3, 3, 5, 1, 1, 4;
%e 9, 3, 2, 5, 1, 5, 1, 1, 4, 3, 2;
%e 10, 1, 5, 1, 1, 3, 8, 2, 1, 1, 1, 2;
%e 11, 3, 1, 5, 6, 4, 2, 4, 3, 1, 1, 1, 7;
%e ...
%Y Cf. A032434, A032435.
%K nonn,tabf
%O 3,4
%A _N. J. A. Sloane_.
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