

A051732


Number of rounds of shuffling required to restore a deck of n cards to its original order: shuffling is done by keeping first card, putting second at end of deck, keeping next, putting next at end and so on.


5



1, 1, 2, 2, 3, 5, 6, 6, 4, 9, 4, 28, 10, 9, 14, 12, 5, 70, 18, 24, 10, 7, 210, 126, 110, 60, 26, 120, 9, 29, 30, 60, 6, 33, 308, 42, 60, 990, 30, 374, 27, 41, 60, 2618, 840, 840, 420, 1386, 24, 15, 50, 644, 840, 53, 18, 1386, 14, 13300, 2520, 1260, 55, 6930, 50, 60, 7
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OFFSET

1,3


COMMENTS

From Andrew Howroyd, Nov 11 2017:(Start)
The shuffling process is the same as the 'deal one, skip one' method described in A289386 except that dealt cards are placed face up. With this variation the first card always remains the first card.
Equivalently, place the numbers 1..n1 on a circle and cyclically mark the 2nd unmarked number until all numbers are marked. The sequence in which the numbers are marked defines a permutation. The order of this permutation is a(n). The numbers 1..n can also be used, but in that case the number 1 should be marked first.
(End)


LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..2000
Gary Antonick, The Polka Dot Puzzle, NY Times Wordplay blog.


FORMULA

a(2^n+1) = n+1.  Ripatti A. (ripatti(AT)inbox.ru), Feb 04 2010
a(A163782(n)+1) = A163782(n).  Andrew Howroyd, Nov 11 2017


EXAMPLE

From Andrew Howroyd, Nov 11 2017: (Start)
a(6) = 5 because it takes 5 rounds of shuffling to return the cards to their original order as illustrated below:
1 2 3 4 5 6
1 3 5 2 6 4
1 5 6 3 4 2
1 6 4 5 2 3
1 4 2 6 3 5
1 2 3 4 5 6
(End)


PROG

(PARI)
P(n, i)={my(d=2*n+12*i); while(d<n, d*=2); 2*nd}
Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
CyclePoly(n, x)={my(q=0); for(i=1, n, my(l=Follow(i, j>P(n, j))); if(l, q+=x^l)); q}
a(n)={my(q=CyclePoly(n, x), m=1); for(i=1, poldegree(q), if(polcoeff(q, i), m=lcm(m, i))); m} \\ Andrew Howroyd, Nov 11 2017


CROSSREFS

Cf. A163782, A289386.
Sequence in context: A137756 A225489 A308773 * A098382 A098180 A270875
Adjacent sequences: A051729 A051730 A051731 * A051733 A051734 A051735


KEYWORD

easy,nonn,nice


AUTHOR

MarieChristine Haton (MarieChristine.Haton(AT)loria.fr)


EXTENSIONS

Name clarified by Andrew Howroyd, Nov 11 2017


STATUS

approved



