A007413 -id:A007413 - cp's OEIS frontend

A354384 Difference sequence of A356133.

2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3

Offset: 1

Clark Kimberling, Aug 04 2022

Cf. A026430, A356133, A091855 (positions of 2), A036554 (positions of 3), A091855 (positions of 4).

u = Accumulate[1 + ThueMorse /@ Range[0, 200]]  (* A026430 *)
v = Complement[Range[Max[u]], u];  (* A356133 *)
Differences[v] (* A354384 *)

a(n) = A007413(n) + 1.

a(n) = A036580(n) + 2.

A173209 Partial sums of A000069.

Original entry on oeis.org

1, 3, 7, 14, 22, 33, 46, 60, 76, 95, 116, 138, 163, 189, 217, 248, 280, 315, 352, 390, 431, 473, 517, 564, 613, 663, 715, 770, 826, 885, 946, 1008, 1072, 1139, 1208, 1278, 1351, 1425, 1501, 1580, 1661, 1743, 1827, 1914, 2002, 2093, 2186, 2280, 2377, 2475, 2575

Offset: 1

Jonathan Vos Post, Feb 12 2010

Partial sums of odious numbers. Second differences give A007413. The subsequence of prime partial sums of odious numbers begins: 3, 7, 163, 431, 613, 2377, 3691, which is a subsequence of A027697. The subsequence of odious partial sums of odious numbers begins: 1, 7, 14, 22, 76, 138, 217, 280, 352, 431, 517, 613, 770, 885.

			a(65) = 1 + 2 + 4 + 7 + 8 + 11 + 13 + 14 + 16 + 19 + 21 + 22 + 25 + 26 + 28 + 31 + 32 + 35 + 37 + 38 + 41 + 42 + 44 + 47 + 49 + 50 + 52 + 55 + 56 + 59 + 61 + 62 + 64 + 67 + 69 + 70 + 73 + 74 + 76 + 79 + 81 + 82 + 84 + 87 + 88 + 91 + 93 + 94 + 97 + 98 + 100 + 103 + 104 + 107 + 109 + 110 + 112 + 115 + 117 + 118 + 121 + 122 + 124 + 127 + 128.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Jean-Paul Allouche, Benoit Cloitre, and Vladimir Shevelev, Beyond odious and evil, arXiv preprint arXiv:1405.6214 [math.NT], 2014.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 52.

Cf. A000069, A000079, A000120, A000773, A000788, A001969, A002042, A002089, A007413, A019568, A023416, A027697, A059015, A083420, A010060, A132679, A133009.

Accumulate@ Select[Range[100], OddQ@ DigitCount[#, 2, 1] &] (* Michael De Vlieger, Nov 01 2022 *)

a(n)=n^2-(n+1)\2+(n%2&&hammingweight(n)%2) \\ Charles R Greathouse IV, Mar 22 2013

a(n) = SUM[i=1..n] A000069(i) = SUM[i=1..n] {i such that A010060(i)=1}.

a(n) = n^2 - n/2 + O(1). - Charles R Greathouse IV, Mar 22 2013

A185311 Keränen's abelian squarefree endomorphism of size 85 on the symbols {1,2,3,4}.

Original entry on oeis.org

1, 2, 3, 1, 3, 4, 3, 2, 3, 4, 3, 1, 4, 3, 4, 2, 4, 1, 2, 1, 3, 1, 2, 1, 4, 2, 1, 2, 3, 2, 4, 2, 3, 2, 1, 3, 2, 3, 4, 3, 1, 3, 2, 1, 2, 4, 1, 2, 1, 3, 1, 4, 3, 2, 3, 4, 3, 1, 3, 4, 2, 3, 2, 1, 3, 2, 3, 4, 3, 1, 3, 4, 3, 2, 4, 3, 4, 1, 4, 2, 4, 3, 2, 3, 1

Offset: 1

N. J. A. Sloane, Jan 25 2012

V. Keränen. Abelian squares are avoidable on 4 letters. In W. Kuich, editor, Proc. ICALP '92, Lecture Notes in Comp. Sci., 623:4152. Springer-Verlag, Berlin, 1992.
V. Keränen. Mathematica in research of avoidable patterns in strings, In V. Keränen and P. Mitic, editors, Mathematics with Vision, Proc. First International Mathematica Symposium (IMS '95, Southampton, England), 259266. Computational Mechanics Publications, 1995.
V. Keränen. The avoidability of regularities in strings (in Finnish). Arkhimedes, 47(1):7179, 1995.
V. Keränen. On abelian repetition-free words. In V. Demidov, editor, Proc. IAS '96, 814. Murmansk State Pedagogical Institute, Murmansk, 1996.
V. Keränen. Repetition-free strings and computer algebra (in Finnish). In C. Gefwert and P. Orponen and J. Seppänen, editors, Logic, Mathematics and the Computer, Proc. Finnish Artificial Intelligence Society, 14:250257. Hakapaino, Helsinki, 1996.
T. Laakso, Musical rendering of an infinite repetition-free string. In C. Gefwert and P. Orponen and J. Seppänen, editors, Logic, Mathematics and the Computer, Proc. Finnish Artificial Intelligence Society, 14:292-297. Hakapaino, Helsinki, 1996.

Arturo Carpi, On the number of abelian square-free words on four letters, Discrete Appl. Math. 81 (1998), no. 1-3, 155-167.
V. Keränen, New Abelian Square-Free DT0L-Languages over 4 Letters

Cf. A007413, A185312.

A317189 A morphic sequence related to the ternary Thue-Morse sequence.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 30 2018

nonn

Michaël Rao, Michel Rigo, Pavel Salimov, Avoiding 2-binomial squares and cubes, arXiv:1310.4743 [cs.FL], 2013.
Michaël Rao, Michel Rigo, Pavel Salimov, Avoiding 2-binomial squares and cubes, Theoretical Computer Science, Volume 572, 23 March 2015, Pages 83-91. See proof of Lemma 1.

Cf. A007413, A036577, A036580, A316826.

Nest[Flatten[# /. {1 -> {2, 0}, 2 -> {1}, 0 -> {2, 1, 0}}] &, {2}, 9 (* must be an odd integer*)] (* Robert G. Wilson v, Jul 30 2018 *)

a(n) = A036577(n), n>0, a(0) = 1. - Michel Dekking, Oct 15 2019

More terms from Robert G. Wilson v, Jul 30 2018

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A354384 Difference sequence of A356133.

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A173209 Partial sums of A000069.

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A185311 Keränen's abelian squarefree endomorphism of size 85 on the symbols {1,2,3,4}.

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A317189 A morphic sequence related to the ternary Thue-Morse sequence.

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