cp's OEIS Frontend

A032435 Triangle of second-to-last man to survive in Josephus problem of n men in a circle with every k-th killed, with 1 <= k <= n and n >= 2.

%I A032435 #44 Jul 08 2025 19:36:28
%S A032435 1,1,2,1,1,3,1,1,2,4,3,2,1,2,5,1,1,5,1,4,6,3,1,2,1,3,4,7,1,4,6,3,1,3,
%T A032435 4,8,3,1,1,2,7,1,3,7,9,5,4,5,3,3,8,1,6,4,10,7,2,9,1,9,4,1,4,3,4,11,1,
%U A032435 5,1,1,3,11,5,1,1,3,2,12,3,8,5,6,9,5,4,10,2,1,1,7,13,5,2,9,2,1,12,7,5
%N A032435 Triangle of second-to-last man to survive in Josephus problem of n men in a circle with every k-th killed, with 1 <= k <= n and n >= 2.
%D A032435 W. W. R. Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover, pp. 32-36, 1987.
%D A032435 M. Kraitchik, "Josephus' Problem", Sec. 3.13 in Mathematical Recreations, New York: W. W. Norton, pp. 93-94, 1942.
%D A032435 Eric W. Weisstein, The CRC Concise Encyclopedia in Mathematics, 2nd ed., Chapman and Hall/CRC, 2002. [The first 8 rows of the triangle appear on p. 1595 of this book under the topic "Josephus Problem".]
%H A032435 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 A032435 Sean A. Irvine, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2020-June/020790.html">A032435 and A032436 Josephus problem data mismatch</a>, message in seqfan, June 2020.
%H A032435 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 second-to-last person to survive in Josephus problem.]
%H A032435 M. Kraitchik, <a href="https://babel.hathitrust.org/cgi/pt?id=wu.89041209552&amp;view=1up&amp;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 A032435 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JosephusProblem.html">Josephus Problem</a>. [It contains a new, apparently corrected, triangle.]
%H A032435 Wikipedia, <a href="https://en.wikipedia.org/wiki/Josephus_problem">Josephus problem</a>.
%H A032435 <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%e A032435 Triangle T(n,k) (with rows n >= 2 and columns k = 2..n) begins
%e A032435    1, 1;
%e A032435    2, 1, 1;
%e A032435    3, 1, 1, 2;
%e A032435    4, 3, 2, 1, 2;
%e A032435    5, 1, 1, 5, 1, 4;
%e A032435    6, 3, 1, 2, 1, 3,  4;
%e A032435    7, 1, 4, 6, 3, 1,  3, 4;
%e A032435    8, 3, 1, 1, 2, 7,  1, 3, 7;
%e A032435    9, 5, 4, 5, 3, 3,  8, 1, 6, 4;
%e A032435   10, 7, 2, 9, 1, 9,  4, 1, 4, 3, 4;
%e A032435   11, 1, 5, 1, 1, 3, 11, 5, 1, 1, 3, 2;
%e A032435   ...
%Y A032435 Cf. A032434, A032436.
%K A032435 nonn,tabf
%O A032435 2,3
%A A032435 _N. J. A. Sloane_