A225232 The number of FO3C2 moves required to restore a packet of n playing cards to its original state (order and orientation).

2, 4, 4, 12, 6, 24, 8, 40, 10, 60, 12, 84, 14, 112, 16, 144, 18, 180, 20, 220, 22, 264, 24, 312, 26, 364, 28, 420, 30, 480, 32, 544, 34, 612, 36, 684, 38, 760, 40, 840, 42, 924, 44, 1012, 46, 1104, 48, 1200, 50, 1300, 52, 1404, 54, 1512, 56, 1624, 58, 1740, 60, 1860, 62, 1984

Offset

3,1

Comments

Each FO3C2 move Flips Over the top 3 cards as a unit and then Cuts 2 cards from the top to bottom. - Mulcahy

References

Colm Mulcahy, Mathematical Card Magic: Fifty-Two New Effects, A K Peters, 2013, chapter 9.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 3..1000

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Formula

Assume n >=3. For odd n we have a(n) = n-1 and for even n we have a(n) = (n-2)n/2. Equivalently, a(2k+1) = 2k and a(2k) = 2k(k-1).

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). - Colin Barker, Jun 04 2014

G.f.: 2*x^3*(x^2-2*x-1) / ((x-1)^3*(x+1)^3). - Colin Barker, Jun 04 2014

Prog

(PARI) a(n)={
    if(n<6, return(if(n>3, 4, 2)));
    n--;
    my(deck=vector(n, i, i), original=deck, steps);
    while(1,
        steps+=2;
        deck=concat(deck[5..n], -[deck[2], deck[1], deck[4], deck[3]]);
        if(deck==original, return(steps))
    )
}; \\ Charles R Greathouse IV, May 03 2013
(PARI) a(n)=if(n%2, n-1, n*(n-2)/2) \\ Charles R Greathouse IV, May 06 2013
(PARI) Vec(2*x^3*(x^2-2*x-1)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Jun 04 2014

Crossrefs

The even numbered terms are A046092.

Cf. A106232.

Sequence in context: A186992 A186993 A186973 * A326824 A319210 A292303

Adjacent sequences: A225229 A225230 A225231 * A225233 A225234 A225235

Keyword

nonn,easy

Author

Colm Mulcahy, May 03 2013

Extensions

a(10), a(12)-a(64) from Charles R Greathouse IV, May 03 2013

Status

approved