

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
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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((n1)/3) or 2*ceiling((n1)/3).  Andrew Howroyd, Apr 27 2020


LINKS



FORMULA

a(3*n+1) = 2*n; a(3*n) = a(3*n1) = 2*n*(n1).  Andrew Howroyd, Apr 27 2020
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(n3)  3*a(n6) + a(n9) for n>12.
(End)


PROG

(PARI) a(n)={2*((n1)\3)*if(n%3==1, 1, (n1)\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



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS

a(16) corrected and terms a(17) and beyond from Andrew Howroyd, Apr 27 2020


STATUS

approved



