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A221564
The number of FO4C3 moves required to restore a packet of n playing cards to its original state (order and orientation), where each move Flips Over the top four (4) as a unit and then Cuts three (3) cards from the top to bottom.
1
2, 4, 4, 4, 12, 12, 6, 24, 24, 8, 40, 40, 10, 60, 60, 12, 84, 84, 14, 112, 112, 16, 144, 144, 18, 180, 180, 20, 220, 220, 22, 264, 264, 24, 312, 312, 26, 364, 364, 28, 420, 420, 30, 480, 480, 32, 544, 544, 34, 612, 612, 36, 684, 684, 38, 760, 760, 40, 840, 840
OFFSET
4,1
COMMENTS
Conjecture: a(3k+1) = 2k.
The top card remains on top but is flipped over with each move. The remaining cards split into three cycles either of length 2*floor((n-1)/3) or 2*ceiling((n-1)/3). - Andrew Howroyd, Apr 27 2020
FORMULA
a(3*n+1) = 2*n; a(3*n) = a(3*n-1) = 2*n*(n-1). - Andrew Howroyd, Apr 27 2020
From Colin Barker, Apr 29 2020: (Start)
G.f.: 2*x^4*(1 + 2*x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x + x^2)^3).
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>12.
(End)
PROG
(PARI) a(n)={2*((n-1)\3)*if(n%3==1, 1, (n-1)\3+1)} \\ Andrew Howroyd, Apr 27 2020
(PARI) Vec(2*x^4*(1 + 2*x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x + x^2)^3) + O(x^40)) \\ Colin Barker, Apr 29 2020
CROSSREFS
Cf. A225232.
Sequence in context: A107058 A332336 A101449 * A134188 A140295 A291780
KEYWORD
nonn,easy
AUTHOR
Colm Mulcahy, May 04 2013
EXTENSIONS
a(16) corrected and terms a(17) and beyond from Andrew Howroyd, Apr 27 2020
STATUS
approved