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A193020 Number of distinct self-shuffles 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

See Jeffrey Shallit's A191755 for the definition of self-shuffle 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^k-5)=3k-6, a(2^k-4)=k*(k-1)/2, a(2^k-3)=2k-2, a(2^k-2)=k, a(2^k-1)=1, a(2^k)=k+1, a(2^k+1)=k^2, a(2^k+2)=2*(k-1)^2, a(2^k+3)=k*(k-1)^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 self-shuffles 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 self-shuffling).  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

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Last modified March 29 04:13 EDT 2020. Contains 333105 sequences. (Running on oeis4.)