A005942 a(2n) = a(n) + a(n+1), a(2n+1) = 2a(n+1), if n >= 2.

1, 2, 4, 6, 10, 12, 16, 20, 22, 24, 28, 32, 36, 40, 42, 44, 46, 48, 52, 56, 60, 64, 68, 72, 76, 80, 82, 84, 86, 88, 90, 92, 94, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182

Offset: 0

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

• nonn
• easy
• nice

Comments

a(n) is the subword complexity (or factor complexity) of Thue-Morse sequence A010060, that is, the number of factors of length n in A010060. See Allouche-Shallit (2003). - N. J. A. Sloane, Jul 10 2012

References

• J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003. See Problem 10, p. 335. - From N. J. A. Sloane, Jul 10 2012
• J. Berstel et al., Combinatorics on Words: Christoffel Words and Repetitions in Words, Amer. Math. Soc., 2008. See p. 83.
• Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• S. Brlek, Enumeration of factors in the Thue-Morse word, Discrete Applied Math. 24 (1989), 83-96.
• A. de Luca and S. Varricchio, Some combinatorial properties of the Thue-Morse sequence and a problem in semigroups, Theoret. Comput. Sci. 63 (1989), 333-348.
• Jakub Konieczny and Clemens Müllner, Arithmetical subword complexity of automatic sequences, arXiv:2309.03180 [math.NT], 2023.
• Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
• Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
• Luís M. S. Russo, Ana D. Correia, Gonzalo Navarro, and Alexandre P. Francisco, Approximating Optimal Bidirectional Macro Schemes, arXiv:2003.02336 [cs.DS], 2020.

Crossrefs

Cf. A005943, A006697, A010060, A275202.

Programs

Haskell

import Data.List (transpose)
a005942 n = a005942_list !! n
a005942_list = 1 : 2 : 4 : 6 : zipWith (+) (drop 6 ts) (drop 5 ts) where
   ts = concat $ transpose [a005942_list, a005942_list]
-- Reinhard Zumkeller, Nov 15 2012

Mathematica

a[0] = 1; a[1] = 2; a[2] = 4; a[3] = 6; a[n_?EvenQ] := a[n] = a[n/2] + a[n/2 + 1]; a[n_?OddQ]  := a[n] = 2*a[(n + 1)/2]; Array[a,60,0] (* Jean-François Alcover, Apr 11 2011 *)

PARI

a(n)=if(n<4,2*max(0,n)+(n==0),if(n%2,2*a((n+1)/2),a(n/2)+a(n/2+1)))

Formula

a(n) = 2*(A006165(n-1) + n - 1), n > 1.

G.f. (1+x^2)/(1-x)^2 + 2*x^2/(1-x)^2 * Sum_{k>=0} (x^2^(k+1)-x^(3*2^k)). - Ralf Stephan, Jun 04 2003

For n > 2, a(n) = 3*(n-1) + A053646(n-1). - Max Alekseyev, May 15 2011

a(n) = 2*A275202(n-1) for n > 1. - N. J. A. Sloane, Jun 04 2019

Extensions

Typo in definition corrected by Reinhard Zumkeller, Nov 15 2012