A145787 Number of times you have to move n cards from one pile to another doing one up, one down, until you obtain the initial sequence.

1, 2, 2, 3, 3, 6, 6, 4, 4, 6, 6, 10, 10, 14, 14, 5, 5, 18, 18, 10, 10, 12, 12, 21, 21, 26, 26, 9, 9, 30, 30, 6, 6, 22, 22, 9, 9, 30, 30, 27, 27, 8, 8, 11, 11, 10, 10, 24, 24, 50, 50, 12, 12, 18, 18, 14, 14, 12, 12, 55, 55, 50, 50, 7, 7, 18, 18, 34, 34, 46, 46

Offset: 1

Author

Hernan Bonsembiante (hernanbon(AT)tutopia.com), Oct 19 2008

Keywords

• nonn

Comments

Let's say you have 3 cards (1 - 2 - 3). You move 1, 2 over 1, 3 below 2. Now you have: (2-1-3). Now you repeat the movement: You move 2, 1 over 2, 3 below 2. Now you have: (1-2-3). The same initial scenario. Total 2 moves. With 4 cards you do it in three moves. For 8 cards you need 4 moves. For 16 cards you need 5 moves. I can assume that for 32 cards I will do it in 6 moves. But for 14 or 15 cards you need 14 moves. I don't know how to predict how many moves for n cards...

Links

• Jon Maiga, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A019567.

Programs

Mathematica

A019567[n_]:=For[m=1,True,m++,If[AnyTrue[{-1,1},Divisible[2^m+#,4n+1]&],Return[m]]]; (* from A019567 *)
Table[A019567[Floor[n/2]],{n,80}] (* Jon Maiga, Oct 06 2019 *)

PARI

deck(n) = {s = "0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ,;:!?.*%$£€=+-&()[]{}_"; v = Vec(s); ss = ""; for (i=1, n, ss = concat(ss, v[i]);); return (ss);}
move(cards) = {v = Vec(cards); s = ""; for (i=1, length(v), if (i % 2, s = concat(s, v[i]), s = concat(v[i], s));); return (s);}
a(n) = {cardsa = deck(n); cardsb = cardsa; diff = 1; nb = 0; while (diff, cardsb = move(cardsb); diff = (cardsa != cardsb); nb++;); return (nb);}
\\ Michel Marcus, Mar 05 2013

Formula

a(n) = A019567(floor(n/2)). - Jon Maiga, Oct 06 2019

Extensions

More terms from Michel Marcus, Mar 05 2013