A227392 Number of 2-Bottom-Card shuffles required to return a deck of size n to its original order.

1, 2, 2, 3, 5, 6, 10, 6, 9, 10, 24, 9, 13, 14, 44, 12, 17, 18, 70, 15, 21, 22, 102, 18, 25, 26, 140, 21, 29, 30, 184, 24, 33, 34, 234, 27, 37, 38, 290, 30, 41, 42, 352, 33, 45, 46, 420, 36, 49, 50

Offset

1,2

Comments

Apply the P card and the B card, the two cards on the bottom of the stack, to do shuffling a deck of cards. The T cards are on the top of cards and the M cards are between the T cards and the P card.

The shuffling steps are like this: (T)(M) P B -> B (T)(M) P -> B (T) P (M).

Let n = T + M +2, (n >= 2). New P = ceiling((n+1)/2).

Count cycles to return original order for the deck of size n.

Order (4k+1)th and (4k+2)th items, k >= 0, can get a Latin square.

Links

Hong-Chang Wang, Table of n, a(n) for n = 1..100

Formula

a(4k+1) = 4k+1, k >= 0;

a(4k+2) = 4k+2, k >= 0;

a(4k+3) = 8k+2, k=0,1;

a(4k+3) = a(4k-1) + 6k + 2, k >= 2;

a(4k+4) = 3*(k+1), k >= 0.

From Joerg Arndt, Jul 16 2013: (Start)

G.f.: x*(2*x^9 + 3*x^8 + 3*x^7 - 4*x^6 - 2*x^4 - 3*x^3 - 2*x^2 - 2*x - 1) / ((x-1)*(x+1)*(x^2+1))^3;

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12). (End)

Example

a(1) = 1
  1
  1
.
a(2) = 2
  1 2
  2 1
  1 2
.
a(3) = 2
  1 2 3
  3 2 1
  1 2 3
.
a(4) = 3
  1 2 3 4
  4 1 3 2
  2 4 3 1
  1 2 3 4
.
a(5) = 5
  1 2 3 4 5
  5 1 4 2 3
  3 5 2 1 4
  4 3 1 5 2
  2 4 5 3 1
  1 2 3 4 5
.
a(6) = 6
  1 2 3 4 5 6
  6 1 2 5 3 4
  4 6 1 3 2 5
  5 4 6 2 1 3
  3 5 4 1 6 2
  2 3 5 6 4 1
  1 2 3 4 5 6
.
a(7) = 10
  1 2 3 4 5 6 7
  7 1 2 6 3 4 5
  5 7 1 4 2 6 3
  3 5 7 6 1 4 2
  2 3 5 4 7 6 1
  1 2 3 6 5 4 7
  7 1 2 4 3 6 5
  5 7 1 6 2 4 3
  3 5 7 4 1 6 2
  2 3 5 6 7 4 1
  1 2 3 4 5 6 7
.
a(8) = 6
  1 2 3 4 5 6 7 8
  8 1 2 3 7 4 5 6
  6 8 1 2 5 3 7 4
  4 6 8 1 7 2 5 3
  3 4 6 8 5 1 7 2
  2 3 4 6 7 8 5 1
  1 2 3 4 5 6 7 8
.
a(9) = 9
  1 2 3 4 5 6 7 8 9
  9 1 2 3 8 4 5 6 7
  7 9 1 2 6 3 8 4 5
  5 7 9 1 4 2 6 3 8
  8 5 7 9 3 1 4 2 6
  6 8 5 7 2 9 3 1 4
  4 6 8 5 1 7 2 9 3
  3 4 6 8 9 5 1 7 2
  2 3 4 6 7 8 9 5 1
  1 2 3 4 5 6 7 8 9

Prog

(C)
int a(int n)
{
    int c[101];
    int i, step, B, P, m;
    for (i=1 ; i<n+1 ; i++)  { c[i] = i; }
    step = 0;
next:
    B = c[n];
    for (i=n-1 ; i>0 ; i--)  { c[i+1] = c[i]; }
    c[1] = B;
    m = (n+2)/2;
    P = c[n];
    if ( n>2 )  { for (i=n-1 ; i>=m ; i--)  c[i+1] = c[i]; }
    c[m] = P;
    step++;
    for (i=1 ; i<n ; i++)  { if ( c[i] != i ) goto next; }
    return step;
}
#include <stdio.h>
int main()
{
    int n;
    for (n=1; n<=100; ++n)  printf("%d, ", a(n));
    printf("\n");
    return 0;
}

Crossrefs

Sequence in context: A291296 A292906 A160235 * A050380 A241652 A241636

Adjacent sequences: A227389 A227390 A227391 * A227393 A227394 A227395

Keyword

nonn

Author

Hong-Chang Wang, Jul 10 2013

Status

approved