A006697 Number of subwords of length n in infinite word generated by a -> aab, b -> b. (Formerly M1001)

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

0,2

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.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A154255 A232739 A342712 * A294860 A183920 A079717

Adjacent sequences: A006694 A006695 A006696 * A006698 A006699 A006700

Keyword

nonn,easy,nice

Author

N. J. A. Sloane and Jeffrey Shallit

Extensions

More terms from Michel ten Voorde, Apr 11 2001

Status

approved