A221564 - OEIS

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.

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

(list; graph; refs; listen; history; text; internal format)

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

LINKS

Andrew Howroyd, Table of n, a(n) for n = 4..1000

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

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