

A193020


Number of distinct selfshuffles of the word given by the binary representation of n.


2



1, 1, 2, 1, 3, 4, 3, 1, 4, 9, 8, 6, 6, 6, 4, 1, 5, 16, 18, 18, 13, 16, 18, 8, 10, 18, 13, 9, 10, 8, 5, 1, 6, 25, 32, 40, 27, 40, 54, 30, 19, 40, 32, 27, 37, 36, 32, 10, 15, 40, 37, 36, 24, 27, 27, 12, 20, 30, 19, 12, 15, 10, 6, 1, 7, 36, 50, 75, 48, 77, 120
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OFFSET

0,3


COMMENTS

See Jeffrey Shallit's A191755 for the definition of selfshuffle and a link to a preprint of the paper "Shuffling and Unshuffling".
An examination of the terms of the sequence leads to the following conjectures (in each case with the caveat that k must exceed a certain lower bound): a(2^k5)=3k6, a(2^k4)=k*(k1)/2, a(2^k3)=2k2, a(2^k2)=k, a(2^k1)=1, a(2^k)=k+1, a(2^k+1)=k^2, a(2^k+2)=2*(k1)^2, a(2^k+3)=k*(k1)^2/2. To illustrate, consider a(2^k+1); we get, for k=1, 2, 3, ..., a(3)=1, a(5)=4, a(9)=9, a(17)=16, a(33)=25, a(65)=36, a(129)=49, a(257)=64,..., leading to the conjecture that a(2^k+1)=k^2. The other conjectures were arrived at in the same manner.


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..2048


EXAMPLE

The binary representation of n=9 is 1001, which has the nine distinct selfshuffles 1'0'0'1001'1, 1'0'0'101'01, 1'0'0'1'1001, 1'0'10'001'1, 1'0'10'01'01, 1'0'10'1'001, 1'10'0'001'1, 1'10'0'01'01, 1'10'0'1'001 (although 1' is identical to 1, and similarly for 0' and 0, the apostrophes indicate one way in which the digits may be assigned to the two copies of the word 1001 and 1'0'0'1' before selfshuffling). Thus a(9)=9.


CROSSREFS

Cf. A191755, A192296.
Sequence in context: A050273 A182511 A187064 * A301471 A237124 A233547
Adjacent sequences: A193017 A193018 A193019 * A193021 A193022 A193023


KEYWORD

nonn


AUTHOR

John W. Layman, Jul 14 2011


STATUS

approved



