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A225232 The number of FO3C2 moves required to restore a packet of n playing cards to its original state (order and orientation).

%I A225232 #29 Feb 12 2024 13:29:14
%S A225232 2,4,4,12,6,24,8,40,10,60,12,84,14,112,16,144,18,180,20,220,22,264,24,
%T A225232 312,26,364,28,420,30,480,32,544,34,612,36,684,38,760,40,840,42,924,
%U A225232 44,1012,46,1104,48,1200,50,1300,52,1404,54,1512,56,1624,58,1740,60,1860,62,1984
%N A225232 The number of FO3C2 moves required to restore a packet of n playing cards to its original state (order and orientation).
%C A225232 Each FO3C2 move Flips Over the top 3 cards as a unit and then Cuts 2 cards from the top to bottom. - Mulcahy
%D A225232 Colm Mulcahy, Mathematical Card Magic: Fifty-Two New Effects, A K Peters, 2013, chapter 9.
%H A225232 Charles R Greathouse IV, <a href="/A225232/b225232.txt">Table of n, a(n) for n = 3..1000</a>
%H A225232 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A225232 Assume n >=3. For odd n we have a(n) = n-1 and for even n we have a(n) = (n-2)n/2. Equivalently, a(2k+1) = 2k and a(2k) = 2k(k-1).
%F A225232 a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). - _Colin Barker_, Jun 04 2014
%F A225232 G.f.: 2*x^3*(x^2-2*x-1) / ((x-1)^3*(x+1)^3). - _Colin Barker_, Jun 04 2014
%o A225232 (PARI) a(n)={
%o A225232     if(n<6,return(if(n>3,4,2)));
%o A225232     n--;
%o A225232     my(deck=vector(n,i,i),original=deck,steps);
%o A225232     while(1,
%o A225232         steps+=2;
%o A225232         deck=concat(deck[5..n],-[deck[2],deck[1],deck[4],deck[3]]);
%o A225232         if(deck==original,return(steps))
%o A225232     )
%o A225232 }; \\ _Charles R Greathouse IV_, May 03 2013
%o A225232 (PARI) a(n)=if(n%2,n-1,n*(n-2)/2) \\ _Charles R Greathouse IV_, May 06 2013
%o A225232 (PARI) Vec(2*x^3*(x^2-2*x-1)/((x-1)^3*(x+1)^3) + O(x^100)) \\ _Colin Barker_, Jun 04 2014
%Y A225232 The even numbered terms are A046092.
%Y A225232 Cf. A106232.
%K A225232 nonn,easy
%O A225232 3,1
%A A225232 _Colm Mulcahy_, May 03 2013
%E A225232 a(10), a(12)-a(64) from _Charles R Greathouse IV_, May 03 2013