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A006355
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Number of binary vectors of length n containing no singletons.
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83
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1, 0, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634
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OFFSET
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0,3
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COMMENTS
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Number of cvtemplates at n-2 letters given <= 2 consecutive consonants or vowels (n >= 4).
Number of (n,2) Freiman-Wyner sequences.
Diagonal sums of the Riordan array ((1-x+x^2)/(1-x), x/(1-x)), A072405 (where this begins 1,0,1,1,1,1,...). - Paul Barry, May 04 2005
Pisano period lengths: 1, 1, 8, 3, 20, 8, 16, 6, 24, 20, 10, 24, 28, 16, 40, 12, 36, 24, 18, 60, ... . - R. J. Mathar, Aug 10 2012
Also the number of matchings in the (n-2)-pan graph for n >= 5. - Eric W. Weisstein, Oct 03 2017
a(n) is the number of bimultus bitstrings of length n. A bitstring is bimultus if each of its 1's possess at least one neighboring 1 and each of its 0's possess at least one neighboring 0. - Steven Finch, May 26 2020
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REFERENCES
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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 16, 51.
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LINKS
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FORMULA
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a(n+2) = F(n-1) + F(n+2), for n > 0.
G.f.: (1-x+x^2)/(1-x-x^2). - Paul Barry, May 04 2005
G.f.: 1 - x + x*Q(0), where Q(k) = 1 + x^2 + (2*k+3)*x - x*(2*k+1 + x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013
a(n) = (2^(-n)*((1-r)^n*(1+r) + (-1+r)*(1+r)^n)) / r for n > 0, where r=sqrt(5). - Colin Barker, Jan 28 2017
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2))/5 - 1. - Stefano Spezia, Apr 18 2022
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EXAMPLE
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a(6)=10 because we have: 000000, 000011, 000111, 001100, 001111, 110000, 110011, 111000, 111100, 111111. - Geoffrey Critzer, Jan 26 2014
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MAPLE
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a:= n-> if n=0 then 1 else (Matrix([[2, -2]]). Matrix([[1, 1], [1, 0]])^n) [1, 1] fi: seq(a(n), n=0..38); # Alois P. Heinz, Aug 18 2008
a := n -> ifelse(n=0, 1, -2*I^n*ChebyshevU(n-2, -I/2)):
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MATHEMATICA
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Join[{1}, Last[#] - First[#] & /@ Partition[Fibonacci[Range[-1, 40]], 4, 1]] (* Harvey P. Dale, Sep 30 2011 *)
CoefficientList[Series[(-1 + x - x^2)/(-1 + x + x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Oct 03 2017 *)
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PROG
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(Haskell)
a006355 n = a006355_list !! n
a006355_list = 1 : fib2s where
fib2s = 0 : map (+ 1) (scanl (+) 1 fib2s)
(Magma) [1] cat [Lucas(n) - Fibonacci(n): n in [1..50]]; // Vincenzo Librandi, Aug 02 2014
(PARI) my(x='x+O('x^50)); Vec((1-x+x^2)/(1-x-x^2)) \\ Altug Alkan, Nov 01 2015
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CROSSREFS
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Except for initial term, = 2*Fibonacci numbers (A000045).
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KEYWORD
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nonn,easy,nice
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AUTHOR
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David M. Bloom
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EXTENSIONS
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STATUS
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approved
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