A080426 - OEIS

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 n-th run of 3's.

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

(list; graph; refs; listen; history; text; internal format)

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 Thue-Morse) with 3.

Length of n-th 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_{n-1}. 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, Anti-Fibonacci numbers, Oct 23 2014.

FORMULA

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

MATHEMATICA

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

nsteps=7; Flatten[SubstitutionSystem[{1->{3}, 3->{1, 1, 3}}, {1}, nsteps]] (* Paolo Xausa, Aug 12 2022, using D. R. Hofstadter's construction *)

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

(PARI)

A080426(nmax) = my(a=[1], s=[[1, 3, 1], [], [1, 3, 3, 3, 1]]); while(length(a)<nmax, a=concat(vecextract(s, a))); a[1..nmax];

A080426(100) \\ Paolo Xausa, Sep 14 2022, using method (1) from comments

(Python)

def A080426(nmax):