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A032434 Triangle read by rows: last survivors of Josephus elimination process. 11
1, 2, 1, 3, 3, 2, 4, 1, 1, 2, 5, 3, 4, 1, 2, 6, 5, 1, 5, 1, 4, 7, 7, 4, 2, 6, 3, 5, 8, 1, 7, 6, 3, 1, 4, 4, 9, 3, 1, 1, 8, 7, 2, 3, 8, 10, 5, 4, 5, 3, 3, 9, 1, 7, 8, 11, 7, 7, 9, 8, 9, 5, 9, 5, 7, 7, 12, 9, 10, 1, 1, 3, 12, 5, 2, 5, 6, 11, 13, 11, 13, 5, 6, 9, 6, 13, 11, 2, 4, 10, 8, 14, 13, 2, 9 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

T(n,k) is the surviving integer under the following elimination process. Arrange 1,2,3,...,n in a circle, increasing clockwise. Starting with i=1, delete the integer k-1 places clockwise from i. Repeat, counting k-1 places from the next undeleted integer, until only one integer remains. - After John W. Layman.

REFERENCES

Ball, W. W. R. and Coxeter, H. S. M., Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 32-36, 1987.

Kraitchik, M. "Josephus' Problem." Sec. 3.13 in Mathematical Recreations. New York: W.W. Norton, pp. 93-94, 1942.

Odlyzko, A. M. and Wilf, H. S. "Functional Iteration and the Josephus Problem." Glasgow Math. J. 33, 235-240, 1991.

LINKS

T. D. Noe, Rows n=1..50, flattened

Ph. Dumas, Algebraic aspects of B-regular series

L. Halbeisen, The Josephus Problem

L. Halbeisen and N. Hungerbuehler, The Josephus Problem

A. M. Odlyzko and H. S. Wilf, Functional iteration and the Josephus problem

Eric Weisstein's World of Mathematics, Josephus Problem.

FORMULA

Recurrence: T(1, k) = 1, T(n, k) = [T(n-1, k)+k] mod n if this is nonzero and n if not.

EXAMPLE

1

2,1

3,3,2

4,1,1,2

5,3,4,1,2

6,5,1,5,1,4

7,7,4,2,6,3,5

MATHEMATICA

t[1, k_] = 1; t[n_, k_] := t[n, k] = If[m = Mod[t[n-1, k] + k, n]; m != 0, m, n]; Flatten[ Table[ t[n, k], {n, 1, 14}, {k, 1, n}]] (* Jean-Fran├žois Alcover, Sep 25 2012 *)

PROG

(PARI) T(n, k)=local(t): if(n<2, n>0, t=(T(n-1, k)+k)%n: if(t, t, n))

CROSSREFS

Cf. A032435, A032436. Second column is A006257, third column is A054995. Diagonal T(n, n) is A007495.

Sequence in context: A102746 A123143 A128133 * A002347 A006642 A210595

Adjacent sequences:  A032431 A032432 A032433 * A032435 A032436 A032437

KEYWORD

nonn,tabl,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Edited by Ralf Stephan, May 18 2004

STATUS

approved

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Last modified April 16 21:07 EDT 2014. Contains 240627 sequences.