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A129715 Number of runs in all Fibonacci binary words of length n. A Fibonacci binary word is a binary word having no 00 subword. A run is a maximal sequence of consecutive identical letters. 3
0, 2, 5, 11, 22, 43, 81, 150, 273, 491, 874, 1543, 2705, 4714, 8173, 14107, 24254, 41555, 70977, 120894, 205401, 348187, 589010, 994511, 1676257, 2820818, 4739861, 7953515, 13328998, 22310971, 37304049, 62307558, 103968225, 173324939 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
a(n) = Sum(k*A129714(n,k), k=0..n).
a(n) = A241701(3n+1,n) for n>0. - Alois P. Heinz, Apr 27 2014
LINKS
FORMULA
G.f.: z(2+z-z^2-z^3)/(1-z-z^2)^2. Rec. rel.: a(n)=a(n-1)+a(n-2)+2F(n) for n>=3, where F(n) is a Fibonacci number (F(0)=0,F(1)=1).
EXAMPLE
a(3)=11 because in the Fibonacci binary words 011, 111, 101, 010 and 110 we have a total of 2+1+3+3+2=11 runs.
MAPLE
with(combinat): a[0]:=0: a[1]:=2: a[2]:=5: for n from 3 to 40 do a[n]:=a[n-1]+a[n-2]+2*fibonacci(n) od: seq(a[n], n=0..40);
MATHEMATICA
CoefficientList[Series[x (2 + x - x^2 - x^3)/(1 - x - x^2)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Apr 28 2014 *)
LinearRecurrence[{2, 1, -2, -1}, {0, 2, 5, 11, 22}, 40] (* Harvey P. Dale, Nov 09 2022 *)
CROSSREFS
Cf. A129714.
Sequence in context: A134508 A091357 A309950 * A024493 A130781 A352045
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 12 2007
STATUS
approved

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Last modified April 22 10:51 EDT 2024. Contains 371897 sequences. (Running on oeis4.)