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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.

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`%I #20 Apr 29 2020 14:40:57
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`%S 2,4,4,4,12,12,6,24,24,8,40,40,10,60,60,12,84,84,14,112,112,16,144,
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`%T 144,18,180,180,20,220,220,22,264,264,24,312,312,26,364,364,28,420,
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`%U 420,30,480,480,32,544,544,34,612,612,36,684,684,38,760,760,40,840,840
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`%N 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.
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`%C Conjecture: a(3k+1) = 2k.
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`%C 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
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`%H Andrew Howroyd, <a href="/A221564/b221564.txt">Table of n, a(n) for n = 4..1000</a>
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`%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1).
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`%F a(3*n+1) = 2*n; a(3*n) = a(3*n-1) = 2*n*(n-1). - _Andrew Howroyd_, Apr 27 2020
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`%F From _Colin Barker_, Apr 29 2020: (Start)
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`%F G.f.: 2*x^4*(1 + 2*x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x + x^2)^3).
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`%F a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>12.
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`%F (End)
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`%o (PARI) a(n)={2*((n-1)\3)*if(n%3==1, 1, (n-1)\3+1)} \\ _Andrew Howroyd_, Apr 27 2020
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`%o (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
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`%Y Cf. A225232.
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`%K nonn,easy
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`%O 4,1
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`%A _Colm Mulcahy_, May 04 2013
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`%E a(16) corrected and terms a(17) and beyond from _Andrew Howroyd_, Apr 27 2020
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