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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A001911 a(n) = Fibonacci(n+3) - 2.

Original entry on oeis.org

0, 1, 3, 6, 11, 19, 32, 53, 87, 142, 231, 375, 608, 985, 1595, 2582, 4179, 6763, 10944, 17709, 28655, 46366, 75023, 121391, 196416, 317809, 514227, 832038, 1346267, 2178307, 3524576, 5702885, 9227463, 14930350, 24157815, 39088167, 63245984
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

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
Ternary words of length n - 1 with subwords (0, 1), (0, 2) and (2, 2) not allowed. - Olivier Gérard, Aug 28 2012
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
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
a(n) is the length of the n-th palindromic prefix of the infinite Fibonacci word A003849. - Dimitri Hendriks, May 19 2014
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
Can be expressed in terms of a rule similar to Recamán's sequence (A005132). Instead of following the Recamán rule "subtract if possible, otherwise add", this sequence follows the rule "If subtraction is possible, do nothing; otherwise add." For example when at the fourth term, 6, it is possible to subtract 4 (giving 2 which is not in {0, 1, 3, 6}) so nothing is done with 4. It is not possible to subtract 5 (6-5=1, which is in {0, 1, 3, 6}), so it is added, resulting in 11. - Matthew Malone, Jan 03 2020
For n>=1, a(n) is the maximum number of vertices (Moore bound) of a (1,1)-regular mixed graph with diameter n-1. - Miquel A. Fiol, Jun 24 2024
Repunits in the lazy Fibonacci representation (A104326), and which is the first row of array A372501. - A.H.M. Smeets, Jun 25 2025

Examples

			G.f. = x + 3*x^2 + 6*x^3 + 11*x^4 + 19*x^5 + 32*x^6 + 53*x^7 + 87*x^8 + ...

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 233.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000032, A000045, A101220, A104326, A108617, A181631, A372501.
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]
Partial sums of A000045(n+1).
Right-hand column 3 of triangle A011794.
See also A165910.
Subsequence of A226538.
Column k=3 of A261019.

Programs

  • Haskell
    a001911 n = a001911_list !! n
a001911_list = 0 : 1 : map (+ 2) (zipWith (+) a001911_list $ tail a001911_list)
-- Reinhard Zumkeller, Jun 18 2013
  • Magma
    [(Fibonacci(n+3))-2: n in [0..85]]; // Vincenzo Librandi, Apr 23 2011
  • Maple
    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
A001911:=(1+z)/(z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation with another offset
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
  • Mathematica
    Table[Fibonacci[n+3] -2, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)
LinearRecurrence[{2,0,-1}, {0,1,3}, 40] (* Harvey P. Dale, Jun 06 2011 *)
Fibonacci[Range[3,40]]-2 (* Harvey P. Dale, Jun 28 2015 *)
  • PARI
    a(n)=fibonacci(n+3)-2 \\ Charles R Greathouse IV, Mar 14 2012
  • SageMath
    [fibonacci(n+3)-2 for n in range(60)] # G. C. Greubel, Oct 21 2024

Formula

From Michael Somos, Jun 09 1999: (Start)
a(n) = A000045(n+3) - 2.
a(n) = a(n-1) + a(n-2) + 2, a(0) = 0, a(1) = 1. (End)
G.f.: x*(1+x)/((1-x)*(1-x-x^2)).
Sum of consecutive pairs of A000071 (partial sums of Fibonacci numbers). - Paul Barry, Apr 17 2004
a(n) = A101220(2, 1, n). - Ross La Haye, Jan 28 2005
a(n) = A108617(n+1, 2) = A108617(n+1, n-1) for n > 0. - Reinhard Zumkeller, Jun 12 2005
a(n) = term (1,1) in the 1 X 3 matrix [0,-1,1].[1,1,0; 1,0,0; 2,0,1]^n. - Alois P. Heinz, Jul 24 2008
a(0) = 0, a(1) = 1, a(2) = 3, a(n) = 2*a(n-1)-a(n-3). - Harvey P. Dale, Jun 06 2011
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
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
a(n) = A000032(6+n)-1 mod A000045(6+n). - Mario C. Enriquez, Apr 01 2017
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 2*exp(x). - Stefano Spezia, May 08 2022

Extensions

More terms and better description from Michael Somos

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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