

A080426


a(1)=1, a(2)=3; all terms are either 1 or 3; each run of 3's is followed by a run of two 1's; and a(n) is the length of the nth run of 3's.


10



1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 3, 3, 1, 1, 3, 3, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 3, 3, 1, 1, 3, 3, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 3, 3, 1, 1, 3, 3, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 3, 3, 1, 1
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OFFSET

1,2


COMMENTS

It appears that the sequence can be calculated by any of the following three methods: (1) Start with 1 and repeatedly replace (simultaneously) all 1's with 1,3,1 and all 3's with 1,3,3,3,1. (Equivalently, trajectory of 1 under the morphism 1 > 1,3,1; 3 > 1,3,3,3,1.  N. J. A. Sloane, Nov 03 2019] (2) a(n)= A026490(2n). (3) Replace each 2 in A026465 (run lengths in ThueMorse) with 3.
Length of nth run of 1's in the Feigenbaum sequence A035263 = 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, ....  Philippe Deléham, Apr 18 2004
Another construction. Let S_0 = 1, and let S_n be obtained by applying the morphism 1 > 3, 3 > 113 to S_{n1}. The sequence is the concatenation S_0, S_1, S_2, ...  D. R. Hofstadter, Oct 23 2014
a(n+1) is the number of times n appears in A003160.  John Keith, Dec 31 2020


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. R. Hofstadter, AntiFibonacci numbers, Oct 23 2014


FORMULA

a(1) = 1; for n>1, a(n) = A003156(n)  A003156(n1).  Philippe Deléham, Apr 16 2004


MATHEMATICA

Position[ Nest[ Flatten[# /. {0 > {0, 2, 1}, 1 > {0}, 2 > {0}}]&, {0}, 8], 0] // Flatten // Differences // Prepend[#, 1]& (* JeanFrançois Alcover, Mar 14 2014, after Philippe Deléham *)


PROG

(Haskell) following Deléham
import Data.List (group)
a080426 n = a080426_list !! n
a080426_list = map length $ filter ((== 1) . head) $ group a035263_list
 Reinhard Zumkeller, Oct 27 2014


CROSSREFS

Cf. A026465, A026490, A035263, A003156, A328979, A003160.
Arises in the analysis of A075326, A249031 and A249032.
Sequence in context: A094782 A035666 A060592 * A230293 A133116 A059959
Adjacent sequences: A080423 A080424 A080425 * A080427 A080428 A080429


KEYWORD

nonn


AUTHOR

John W. Layman, Feb 18 2003


STATUS

approved



