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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
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
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
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MATHEMATICA
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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