

A001387


The binary "look and say" sequence.


8



1, 11, 101, 111011, 11110101, 100110111011, 111001011011110101, 111100111010110100110111011, 100110011110111010110111001011011110101, 1110010110010011011110111010110111100111010110100110111011
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OFFSET

1,2


COMMENTS

I conjecture that the ratio r(n) of the number of "1"s to the number of "0"s in a(n) converges to 5/3 (or some nearby limit).  Joseph L. Pe, Jan 31 2003
The ratio r(n) of the number of "1"s to the number of "0"s in a(n) actually converges to ((101  10*sqrt(93))*a^2 + (139  13*sqrt(93))*a  76)/108, where a = (116 + 12*sqrt(93))^(1/3). This ratio has decimal expansion 1.6657272222676...  Nathaniel Johnston, Nov 07 2010 [Corrected by Kevin J. Gomez, Dec 12 2017]
Reading terms as binary numbers and converting to decimal gives A049190.  Andrey Zabolotskiy, Dec 12 2017


LINKS

John Cerkan, Table of n, a(n) for n = 1..17
J. H. Conway, The weird and wonderful chemistry of audioactive decay, Eureka 46 (1986) 516, reprinted in: Open Problems in Communications and Computations, Springer, 1987, 173188.
Nathaniel Johnston, The Binary "LookandSay" Sequence
Thomas Morrill, Look, Knave, arXiv:2004.06414 [math.CO], 2020.
Torsten Sillke, The binary form of Conway's sequence


EXAMPLE

To get the 5th term, for example, note that 4th term has three (11 in binary!) 1's, one (1) 0 and two (10) 1's, giving 11 1 1 0 10 1.


MATHEMATICA

a[1] := 1; a[n_] := a[n] = FromDigits[Flatten[{IntegerDigits[Length[#], 2], First[#]}& /@ Split[IntegerDigits[a[n1]]]]]; Map[a, Range[20]] (* Peter J. C. Moses, Mar 24 2013 *)
Nest[Append[#, FromDigits@ Flatten@ Map[Reverse /@ IntegerDigits[Tally@ #, 2] &, Split@ IntegerDigits@ Last@ #]] &, {1}, 9] (* Michael De Vlieger, Dec 12 2017 *)


CROSSREFS

Cf. A005150, A001391, A049190, A049194.
Sequence in context: A156668 A103992 A185949 * A247863 A180280 A100580
Adjacent sequences: A001384 A001385 A001386 * A001388 A001389 A001390


KEYWORD

nonn,base


AUTHOR

Thomas L. York


EXTENSIONS

New name from Andrey Zabolotskiy, Dec 13 2017


STATUS

approved



