A321580 Numbers k such that it is possible to reverse a deck of k cards by a sequence of perfect Faro shuffles with cut.

1, 2, 4, 8, 10, 12, 16, 18, 24, 26, 28, 32, 36, 40, 42, 52, 56, 58, 60, 64, 66, 80, 82, 96, 98, 100, 106, 108, 112, 120, 124, 128, 130, 136, 138, 144, 148, 156, 162, 168, 170, 172, 176, 178, 180, 184, 192, 196, 200, 204, 208, 210, 226, 228, 240, 242, 248, 250

Offset

1,2

Comments

Except for 1, it isn't possible to shuffle backwards an odd number of cards.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..2000

Wikipedia, Faro shuffle

Example

For a deck of 4 cards we'll have the following sequence of shuffles: 1234, 2413, 4321, 3142, 1234. Observe that the reverse order (4321) of 1234 appears in the sequence of shuffles.
For a deck of 5 cards: 12345, 24135, 43215, 31425, 12345. Observe that the reverse order (54321) of 12345 does not appear in the sequence of shuffles.

Prog

(Python)
for n in range(1, 501):
    cards = [i for i in range(1, n + 1)]
    reverse = cards[::-1]
    shuffled = cards.copy()
    reversein = False
    for i in range(n):
        evens = shuffled[1::2]
        odds = shuffled[0::2]
        shuffled = evens + odds
        if shuffled == reverse:
            reversein = True
            print(n, end=", ")
            break
(PARI)
shuffle(v)={my(h=#v\2); vector(#v, i, if(i<=h, 2*i, 2*(i-h)-1))}
permcycs(v)={my(f=vector(#v), L=List()); for(i=1, #v, if(!f[i], my(T=List(), j=i); while(!f[j], f[j]=1; listput(T, j); j=v[j]); listput(L, Vec(T)))); Vec(L)}
ok(n)={my(v=permcycs(shuffle([1..n])), e=-1); for(k=1, #v, my(p=v[k]); if(#p>1||n%2==0||2*p[1]<>n+1, my(h=#p\2); if(e<0, e=valuation(#p, 2)); if(valuation(#p, 2)<>e || e==0 || p[1..h]+p[h+1..2*h]<>vector(h, i, n+1), return(0)))); 1} \\ Andrew Howroyd, Nov 13 2018

Crossrefs

Cf. A024222, A071642, A321512.

Sequence in context: A036990 A097498 A346502 * A224694 A140900 A166936

Adjacent sequences: A321577 A321578 A321579 * A321581 A321582 A321583

Keyword

nonn

Author

Pedro Menezes, Nov 13 2018

Status

approved