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A006697
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Number of subwords of length n in infinite word generated by a -> aab, b -> b.
(Formerly M1001)
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7
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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; internal format)
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OFFSET
| 0,2
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REFERENCES
| J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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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 (ralf(AT)ark.in-berlin.de), Mar 05 2004
Conjectures: partial sums of A103354, also equal to A094913(n) + 1. - Vladeta Jovovic (vladeta(AT)eunet.rs), 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 (Maximilian.Hasler(AT)gmail.com), Dec 14 2007
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MATHEMATICA
| A103354[n_] := Floor[ FullSimplify[ ProductLog[ 2^n*Log[2]]/Log[2]]]; Accumulate[ Table[ A103354[n], {n, 1, 54}]] (* From Jean-François Alcover, Dec 13 2011, after M. F. Hasler *)
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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 (Maximilian.Hasler(AT)gmail.com), Dec 14 2007
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CROSSREFS
| Cf. A005943, A005942, A094913, A134457, A134466.
Sequence in context: A022331 A087483 A154255 * A183920 A079717 A175780
Adjacent sequences: A006694 A006695 A006696 * A006698 A006699 A006700
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jeffrey Shallit
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EXTENSIONS
| More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org) Apr 11 2001
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