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A006355 Number of binary vectors of length n containing no singletons. 82
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

a(n) = A119457(n-1,n-2) for n>2. - Reinhard Zumkeller, May 20 2006

Central terms of the triangle in A094570. - Reinhard Zumkeller, Mar 22 2011

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

REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 16, 51.

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 0..4786 (next term has 1001 digits)

Ian F. Blake, The enumeration of certain run length sequences, Information and Control, 55 (1982), 222-237.

A. Burstein, S. Kitaev, T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.

Enoch Haga, Room for Expansion, Word Ways, 33 (No. 2, 2000), pp. 106-113 (see p. 110).

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 898

Sergey Kitaev, Jeffrey Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], 2013.

Index entries for linear recurrences with constant coefficients, signature (1,1).

FORMULA

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

a(n) = 2*F(n-1) for n>0, F(n)=A000045(n) and a(0)=1. - Mircea Merca, Jun 28 2012

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) = A118658(n)-0^n. - M. F. Hasler, Nov 05 2014

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

EXAMPLE

a(6)=10 because we have: 000000, 000011, 000111, 001100, 001111, 110000, 110011, 111000, 111100, 111111. - Geoffrey Critzer, Jan 26 2014

MAPLE

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

MATHEMATICA

lst={1}; Do[AppendTo[lst, Fibonacci[n+3]-Fibonacci[n]], {n, -1, 4*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 11 2009 *)

lst={1}; a=2; s=3; Do[a=s-(a+1); AppendTo[lst, a]; s+=a, {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

Join[{1}, Last[#] - First[#] & /@ Partition[Fibonacci[Range[-1, 40]], 4, 1]] (* Harvey P. Dale, Sep 30 2011 *)

PROG

(PARI) a(n)=if(n, 2*fibonacci(n-1), 1) \\ Charles R Greathouse IV, Mar 14 2012

(Haskell)

a006355 n = a006355_list !! n

a006355_list = 1 : fib2s where

   fib2s = 0 : map (+ 1) (scanl (+) 1 fib2s)

-- Reinhard Zumkeller, Mar 20 2013

(MAGMA) [1] cat [Lucas(n) - Fibonacci(n): n in [1..50]]; // Vincenzo Librandi, Aug 02 2014

(PARI) x='x+O('x^50); Vec((1-x+x^2)/(1-x-x^2)) \\ Altug Alkan, Nov 01 2015

CROSSREFS

Except for initial term, = 2*Fibonacci numbers (A000045).

Essentially the same as A055389, A047992, A068922, A054886 and A090991.

Cf. A097925, A097926. Column 2 in A265584.

Sequence in context: A050194 A228807 A262258 * A055389 A163733 A198834

Adjacent sequences:  A006352 A006353 A006354 * A006356 A006357 A006358

KEYWORD

nonn,easy,nice

AUTHOR

David M. Bloom

EXTENSIONS

More terms from Reinhard Zumkeller, May 20 2006

Corrected by T. D. Noe, Oct 31 2006

STATUS

approved

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Last modified August 17 21:11 EDT 2017. Contains 290656 sequences.