A006697 Number of subwords of length n in infinite word generated by a -> aab, b -> b.

1, 2, 4, 6, 9, 13, 17, 22, 28, 35, 43, 51, 60, 70, 81, 93, 106, 120, 135, 151, 167, 184, 202, 221, 241, 262, 284, 307, 331, 356, 382, 409, 437, 466, 496, 527, 559, 591, 624, 658, 693, 729, 766, 804, 843, 883, 924, 966, 1009, 1053, 1098, 1144, 1191, 1239

Offset: 0

Author

N. J. A. Sloane and Jeffrey Shallit

Keywords

• nonn
• easy
• nice

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
• J.-P. Allouche and J. Shallit, On the subword complexity of the fixed point of a -> aab, b -> b, and generalizations, arXiv preprint arXiv:1605.02361 [math.CO], 2016.
• Gabriele Fici, The Shortest Interesting Binary Words, arXiv:2412.21145 [math.CO], 2024. See p. 12.

Crossrefs

Cf. A005943, A005942, A094913, A134457, A134466.

Programs

Mathematica

A103354[n_] := Floor[ FullSimplify[ ProductLog[ 2^n*Log[2]]/Log[2]]]; Accumulate[ Table[ A103354[n], {n, 1, 54}]] (* Jean-François Alcover, Dec 13 2011, after M. F. Hasler *)

PARI

LambertW(y) = solve( X=1,log(y), X*exp(X)-y)
A006697(n,b=2)=local(m=floor(n+1-LambertW(b^(n+1)*log(b))/log(b)));(b^(m+1)-1)/(b-1)+(n-m)*(n-m+1)/2 \\ M. F. Hasler, Dec 14 2007

Formula

G.f.: 1 + 1/(1-x) + 1/(1-x)^2 * [1/(1-x) - sum(k>=1, x^(2^k+k-1))] (conjectured). - Ralf Stephan, Mar 05 2004

Conjectures: partial sums of A103354, also equal to A094913(n) + 1. - Vladeta Jovovic, Sep 19 2005

a(n) = sum(k=0,n,min(2^k,n-k+1)) = 2^(m+1)-1 + (n-m)(n-m+1)/2 with m = [ n+1-LambertW( 2^(n+1) * log(2) ) / log(2) ] = integer part of the solution to 2^m = n+1-m. (conjectured). - M. F. Hasler, Dec 14 2007

Extensions

More terms from Michel ten Voorde, Apr 11 2001