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A225232 The number of FO3C2 moves required to restore a packet of n playing cards to its original state (order and orientation). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 28 08:27 EST 2020. Contains 332322 sequences. (Running on oeis4.)