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A209232 a(n) is 2^n times the expected value of the shortest run of 0's in a binary word of length n. 2
0, 1, 4, 11, 25, 52, 103, 199, 380, 724, 1382, 2649, 5103, 9881, 19224, 37559, 73646, 144848, 285623, 564429, 1117396, 2215436, 4398054, 8740266, 17385207, 34607218, 68934319, 137386725, 273942683, 546450648, 1090419638 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
a(n) is also the sum of the number of binary words containing at least one 0 and having every consecutive run of 0's of length >= i for i >= 1. In other words, a(n) = A000225(n) + A077855(n) + A130578(n) + A209231(n) + ...
REFERENCES
R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, Chapter 7.
LINKS
FORMULA
O.g.f.: Sum_{k >= 1} (x^k/(1 - x) + 1) / ((1 - x^(k + 1)/(1 - x)^2)) * 1/(1 - x) - 1/(1 - x).
EXAMPLE
a(3) = 11. To the length 3 binary words {0, 0, 0}, {0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {1, 0, 0}, {1, 0, 1}, {1, 1, 0}, {1, 1, 1} we have respectively shortest zero runs of length 3 + 2 + 1 + 1 + 2 + 1 + 1 + 0 = 11.
MATHEMATICA
nn = 30; Apply[Plus, Table[a = x^n/(1 - x); CoefficientList[Series[(a + 1)/((1 - a x/(1 - x)))*1/(1 - x) - 1/(1 - x), {x, 0, nn}], x], {n, 1, nn}]]
CROSSREFS
Cf. A119706.
Sequence in context: A356619 A014173 A290986 * A266337 A262158 A156127
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Jan 12 2013
STATUS
approved

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Last modified April 24 04:14 EDT 2024. Contains 371918 sequences. (Running on oeis4.)