A026600 -id:A026600 - cp's OEIS frontend

A026611 Number of 3's between n-th 2 and (n+1)st 2 in A026600.

1, 2, 0, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 0, 2, 0, 1, 2, 0, 2, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 2, 0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 2, 0, 2, 0, 2, 0, 1, 1, 1, 2, 1

Offset: 1

Clark Kimberling

nonn

A026612 a(n) = number of 1's between n-th 3 and (n+1)st 3 in A026600.

Original entry on oeis.org

0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 2, 0, 1, 2, 0, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 0, 2, 0, 1, 2, 0, 2, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 1, 1, 2, 0, 2, 0, 2, 0, 1, 1

Offset: 1

Clark Kimberling

nonn

A026614 a(n) least k > a(n-1) such that a(k)=s(n), for n >= 2, where s = A026600.

Original entry on oeis.org

1, 6, 9, 11, 15, 18, 20, 24, 27, 28, 33, 36, 38, 42, 45, 47, 51, 54, 55, 60, 63, 65, 69, 72, 74, 78, 81, 84, 87, 90, 92, 96, 99, 101, 105, 108, 109, 114, 117, 119, 123, 126, 128, 132, 135, 136, 141, 144, 146, 150, 153, 155, 159, 162

Offset: 1

Clark Kimberling

nonn

A026606 [1->null]-transform of three-symbol Thue-Morse A026600, with 1 subtracted.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2

Offset: 1

Clark Kimberling

nonn

Old name was: a(n) = b(n)-1, where b(n) = n-th term of A026600 that is not a 1.
From Michel Dekking, Apr 18 2019: (Start)
This sequence is a morphic sequence, i.e., a letter-to-letter projection of a fixed point of a morphism. Let the morphism sigma be given by
    1->123, 2->456, 3->345,4->612, 5->561, 6->234,
and let the letter-to-letter map delta be given by
    1->1, 2->2, 3->1, 4->2, 5->2, 6->1.
Then (a(n)) = delta(x), where x = 1234... is a fixed point of sigma.
This representation can be obtained by noting that this sequence, with 1 added, can also be viewed as the [1->23, 2->23, 3->32]-transform of A026600, and by doubling 1,2 and 3, renaming the resulting six letters as 1,2,3,4,5,6.
(End)

Cf. A026605, A057215.

Name changed by Michel Dekking, Apr 18 2019

A026607 Delete all 2's from A026600 and then replace each 3 with 2.

Original entry on oeis.org

1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2

Offset: 1

Clark Kimberling

nonn

A053838 a(n) = (sum of digits of n written in base 3) modulo 3.

Original entry on oeis.org

0, 1, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 2, 0, 0, 1, 2, 1, 2, 0, 2, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 0, 1, 2, 1, 2, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 0, 1, 2, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 2, 0, 0, 1, 2, 1, 2, 0

Offset: 0

Henry Bottomley, Mar 28 2000

Start with 0, repeatedly apply the morphism 0->012, 1->120, 2->201. This is a ternary version of the Thue-Morse sequence A010060. See Brlek (1989). - N. J. A. Sloane, Jul 10 2012

A090193 is generated by the same mapping starting with 1. A090239 is generated by the same mapping starting with 2. - Andrey Zabolotskiy, May 04 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
S. Brlek, Enumeration of factors in the Thue-Morse word, Discrete Applied Math. 24 (1989), 83-96.
Arthur Dolgopolov, Equitable Sequencing and Allocation Under Uncertainty, Preprint, 2016.
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
Michael Gilleland, Some Self-Similar Integer Sequences
Michel Rigo, Relations on words, arXiv preprint arXiv:1602.03364 [cs.FL], 2016. See Example 17.
Robert Walker, Self Similar Sloth Canon Number Sequences
Index entries for sequences that are fixed points of mappings

Cf. A004128, A010060, A053837, A053839-A053844.

Equals A026600(n+1) - 1.

A053838 := proc(n)
    add(d,d=convert(n,base,3)) ;
    modp(%,3) ;
end proc:
seq(A053838(n),n=0..100) ; # R. J. Mathar, Nov 04 2017

Nest[ Flatten[ # /. {0 -> {0, 1, 2}, 1 -> {1, 2, 0}, 2 -> {2, 0, 1}}] &, {0}, 7] (* Robert G. Wilson v, Mar 08 2005 *)

a(n) = vecsum(digits(n, 3)) % 3; \\ Michel Marcus, May 04 2016

from sympy.ntheory import digits
def A053838(n): return sum(digits(n,3)[1:])%3 # Chai Wah Wu, Feb 28 2025

a(n) = A010872(A053735(n)) =(n+a(floor[n/3])) mod 3. So one can construct sequence by starting with 0 and mapping 0->012, 1->120 and 2->201 (e.g. 0, 012, 012120201, 012120201120201012201012120, ...) and looking at n-th digit of a term with sufficient digits.

a(n) = A004128(n) mod 3. [Gary W. Adamson, Aug 24 2008]

cp's OEIS Frontend

A026611 Number of 3's between n-th 2 and (n+1)st 2 in A026600.

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A026612 a(n) = number of 1's between n-th 3 and (n+1)st 3 in A026600.

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A026614 a(n) least k > a(n-1) such that a(k)=s(n), for n >= 2, where s = A026600.

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A026606 [1->null]-transform of three-symbol Thue-Morse A026600, with 1 subtracted.

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A026607 Delete all 2's from A026600 and then replace each 3 with 2.

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A053838 a(n) = (sum of digits of n written in base 3) modulo 3.

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