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A001911 a(n) = Fibonacci(n+3) - 2.
(Formerly M2546 N1007)
61

%I M2546 N1007

%S 0,1,3,6,11,19,32,53,87,142,231,375,608,985,1595,2582,4179,6763,10944,

%T 17709,28655,46366,75023,121391,196416,317809,514227,832038,1346267,

%U 2178307,3524576,5702885,9227463,14930350,24157815,39088167,63245984

%N a(n) = Fibonacci(n+3) - 2.

%C This is the sequence A(0,1;1,1;2) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - _Wolfdieter Lang_, Oct 17 2010

%C Ternary words of length n - 1 with subwords (0, 1), (0, 2) and (2, 2) not allowed. - _Olivier Gérard_, Aug 28 2012

%C For subsets of (1, 2, 3, 5, 8, 13,...) Fibonacci Maximal terms (Cf. A181631) equals the number of leading 1's per subset. For example, (7-11) in Fibonacci Maximal = (1010, 1011, 1101, 1110, 1111), numbers of leading 1's = (1 + 1 + 2 + 3 + 4) = 11 = a(4) = row 4 of triangle A181631. - _Gary W. Adamson_, Nov 02 2010

%C As in our 2009 paper, we use two types of Fibonacci trees: - Ta: Fibonacci analog of binomial trees; Tb: Binary Fibonacci trees. Let D(r(k)) be the sum over all distances of the form d(r, x), across all vertices x of the tree rooted at r of order k. Ignoring r, but overloading, let D(a(k)) and D(b(k)) be the distance sums for the Fibonacci trees Ta and Tb respectively of the order k. Using the sum-of-product form in Equations (18) and (21) in our paper it follows that F(k+4) - 2 = D(a(k+1)) - D(b(k-1)). - _K.V.Iyer_ and P. Venkata Subba Reddy, Apr 30 2011

%C a(n) is the length of the n-th palindromic prefix of the infinite Fibonacci word A003849. - _Dimitri Hendriks_, May 19 2014

%C The first k terms of the infinite Fibonacci word A014675 are palindromic if and only if k is a positive term of this sequence. - _Clark Kimberling_, Jul 14 2014

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 233.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Charles R Greathouse IV, <a href="/A001911/b001911.txt">Table of n, a(n) for n = 0..4783</a> (next term has 1001 digits)

%H D. J. Broadhurst, <a href="http://arXiv.org/abs/hep-th/9604128">On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory</a>, arXiv:hep-th/9604128, 1996.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H M. Rigo, P. Salimov, and E. Vandomme, <a href="http://www.emis.de/journals/JIS/VOL16/Rigo/rigo3.html">Some Properties of Abelian Return Words</a>, Journal of Integer Sequences, Vol. 16 (2013), #13.2.5.

%H D. G. Rogers, <a href="http://dx.doi.org/10.1007/BFb0102693">An application of renewal sequences to the dimer problem</a>, pp. 142-153 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979.

%H K. Viswanathan Iyer and K. R. Uday Kumar Reddy, <a href="http://arXiv.org/abs/0910.4432">Wiener index of binomial trees and Fibonacci trees</a>, arXiv:0910.4432 [cs.DM], 2009. (Corrigendum: Eq.(23) to be corrected as follows on the right-side: in the fourth term F(k)-1 should be replaced by F(k); a term F(k)*F(K+1)-1 is to be included; pointed out by Emeric Deutsch).

%H Wolfdieter Lang, <a href="/A001911/a001911.pdf">Notes on certain inhomogeneous three term recurrences.</a> [Wolfdieter Lang, Oct 17 2010]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).

%F a(n) = a(n-1) + a(n-2) + 2, a(0) = 0, a(1) = 1. - _Michael Somos_, Jun 09 1999

%F G.f.: x*(1+x)/((1-x)*(1-x-x^2)).

%F Sum of consecutive pairs of A000071 (partial sums of Fibonacci numbers). - _Paul Barry_, Apr 17 2004

%F a(n) = A101220(2, 1, n). - _Ross La Haye_, Jan 28 2005

%F a(n) = A108617(n+1, 2) = A108617(n+1, n-1) for n > 0. - _Reinhard Zumkeller_, Jun 12 2005

%F a(n) = term (1,1) in the 1x3 matrix [0,-1,1].[1,1,0; 1,0,0; 2,0,1]^n. - _Alois P. Heinz_, Jul 24 2008

%F a(0) = 0, a(1) = 1, a(2) = 3, a(n) = 2*a(n-1)-a(n-3). - _Harvey P. Dale_, Jun 06 2011

%F Eigensequence of an infinite lower triangular matrix with the natural numbers as the left border and (1, 0, 1, 0,...) in all other columns. - _Gary W. Adamson_, Jan 30 2012

%F a(n) = (-2+(2^(-n)*((1-sqrt(5))^n*(-2+sqrt(5))+(1+sqrt(5))^n*(2+sqrt(5))))/sqrt(5)). - _Colin Barker_, May 12 2016

%e x + 3*x^2 + 6*x^3 + 11*x^4 + 19*x^5 + 32*x^6 + 53*x^7 + 87*x^8 + ...

%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2]+2 od: seq(a[n],n=0..50); # Miklos Kristof, Mar 09 2005

%p A001911:=(1+z)/(z-1)/(z**2+z-1); # _Simon Plouffe_ in his 1992 dissertation with another offset

%p a:= n-> (Matrix([[0,-1,1]]). Matrix([[1,1,0], [1,0,0], [2,0,1]])^n)[1,1]: seq(a(n), n=0..50); # _Alois P. Heinz_, Jul 24 2008

%t Table[Fibonacci[n + 3] - 2, {n, 0, 5!}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 19 2010 *)

%t LinearRecurrence[{2, 0, -1}, {0, 1, 3}, 40] (* _Harvey P. Dale_, Jun 06 2011 *)

%t Fibonacci[Range[3,40]]-2 (* _Harvey P. Dale_, Jun 28 2015 *)

%o (MAGMA) [(Fibonacci(n+3))-2: n in [0..85]]; // _Vincenzo Librandi_, Apr 23 2011

%o (PARI) a(n)=fibonacci(n+3)-2 \\ _Charles R Greathouse IV_, Mar 14 2012

%o (Haskell)

%o a001911 n = a001911_list !! n

%o a001911_list = 0 : 1 : map (+ 2) (zipWith (+) a001911_list $ tail a001911_list)

%o -- _Reinhard Zumkeller_, Jun 18 2013

%Y a(n) = A000045(n+3) - 2. - _Michael Somos_, Jun 09 1999

%Y Partial sums of F(n+1) = A000045(n+1).

%Y Right-hand column 3 of triangle A011794.

%Y Cf. A001611, A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616. [Added by _N. J. A. Sloane_, Jun 25 2010 in response to a comment from _Aviezri S. Fraenkel_]

%Y Cf. A181631. - _Gary W. Adamson_, Nov 02 2010

%Y See also A165910.

%Y Subsequence of A226538.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E More terms and better description from _Michael Somos_

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Last modified December 8 22:51 EST 2016. Contains 278957 sequences.