A221564 - OEIS

%I #20 Apr 29 2020 14:40:57

%S 2,4,4,4,12,12,6,24,24,8,40,40,10,60,60,12,84,84,14,112,112,16,144,

%T 144,18,180,180,20,220,220,22,264,264,24,312,312,26,364,364,28,420,

%U 420,30,480,480,32,544,544,34,612,612,36,684,684,38,760,760,40,840,840

%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.

%C Conjecture: a(3k+1) = 2k.

%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

%H Andrew Howroyd, <a href="/A221564/b221564.txt">Table of n, a(n) for n = 4..1000</a>

%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).

%F a(3*n+1) = 2*n; a(3*n) = a(3*n-1) = 2*n*(n-1). - _Andrew Howroyd_, Apr 27 2020

%F From _Colin Barker_, Apr 29 2020: (Start)

%F G.f.: 2*x^4*(1 + 2*x + 2*x^2 - x^3) / ((1 - x)^3*(1 + x + x^2)^3).

%F a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>12.

%F (End)

%o (PARI) a(n)={2*((n-1)\3)*if(n%3==1, 1, (n-1)\3+1)} \\ _Andrew Howroyd_, Apr 27 2020

%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

%Y Cf. A225232.

%K nonn,easy

%O 4,1

%A _Colm Mulcahy_, May 04 2013

%E a(16) corrected and terms a(17) and beyond from _Andrew Howroyd_, Apr 27 2020