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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. 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 (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
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):
a, s = "1", "".maketrans({"1":"131", "3":"13331"})
while len(a) < nmax: a = a.translate(s)
return list(map(int, a[:nmax]))
print(A080426(100)) # Paolo Xausa, Aug 30 2022, using method (1) from comments
CROSSREFS
Arises in the analysis of A075326, A249031 and A249032.
Sequence in context: A094782 A035666 A060592 * A230293 A133116 A059959
KEYWORD
nonn
AUTHOR
John W. Layman, Feb 18 2003
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)