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 A006697 Number of subwords of length n in infinite word generated by a -> aab, b -> b. (Formerly M1001) 7
 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 (list; graph; refs; listen; history; text; internal format)
 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, 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. 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: A087483 A154255 A232739 * A294860 A183920 A079717 Adjacent sequences:  A006694 A006695 A006696 * A006698 A006699 A006700 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from Michel ten Voorde Apr 11 2001 STATUS approved

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Last modified October 17 21:37 EDT 2019. Contains 328134 sequences. (Running on oeis4.)