A001611 a(n) = Fibonacci(n) + 1.

1, 2, 2, 3, 4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• hear

Comments

a(0) = 1, a(1) = 2 then the largest number such that a triangle is constructible with three successive terms as sides. - Amarnath Murthy, Jun 03 2003

a(n+2) = A^(n)B(1), n>=0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 2=`0`, 3=`10`, 4=`110`, 6=`1110`, ..., in Wythoff code.

The first-difference sequence is the Fibonacci sequence (A000045). - Roland Schroeder (florola(AT)gmx.de), Aug 05 2010

2 and 3 are the only primes in this sequence.

a(n) is the number of 1 X n nonogram puzzles which can be solved uniquely. See A242876 for puzzle definition. - Lior Manor, Jan 23 2022

References

• G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
• 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

• Vincenzo Librandi, Table of n, a(n) for n = 0..250
• Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
• K.-W. Chen, Greatest Common Divisors in Shifted Fibonacci Sequences, J. Int. Seq. 14 (2011) # 11.4.7.
• Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).
• Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), #15.11.8.
• R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
• Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 5.1.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 402
• Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 97.
• N. S. Mendelsohn, Permutations with confined displacement, Canad. Math. Bull., 4 (1961), 29-38.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Cf. A000045, A097280, A097281.

Cf. 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]

Cf. A002062, A019863, A160536, A212272, A242876.

Programs

Haskell

a001611 = (+ 1) . a000045
a001611_list = 1 : 2 : map (subtract 1)
                       (zipWith (+) a001611_list $ tail a001611_list)
-- Reinhard Zumkeller, Jul 30 2013

Magma

[Fibonacci(n)+1: n in [1..37]]; // Bruno Berselli, Jul 26 2011

Maple

A001611:=-(-1+2*z**2)/(z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation
with(combinat): seq((fibonacci(n)+1), n=0..35);

Mathematica

a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n-2] + a[n-1] - 1; Table[ a[n], {n, 0, 40} ]
Fibonacci[Range[0,50]]+1  (* Harvey P. Dale, Mar 23 2011 *)

PARI

a(n)=fibonacci(n)+1 \\ Charles R Greathouse IV, Jul 25 2011

Formula

G.f.: (1-2*x^2)/(1-2*x+x^3).

a(n) = 2*a(n-1) - a(n-3). - Tanya Khovanova, Jul 13 2007

a(0) = 1, a(1) = 2, a(n) = a(n - 2) + a(n - 1) - 1.

F(4*n) + 1 = F(2*n-1)*L(2*n+1); F(4*n+1) + 1 = F(2*n+1)*L(2*n); F(4*n+2) + 1 = F(2*n+2)*L(2*n); F(4*n+3) + 1 = F(2*n+1)*L(2*n+2) where F(n)=Fibonacci(n) and L(n)=Lucas(n). - R. K. Guy, Feb 27 2003

a(1) = 2; a(n+1)=floor(a(n)*(sqrt(5)+1)/2). - Roland Schroeder (florola(AT)gmx.de), Aug 05 2010

a(n) = Sum_{k=0..n+1} Fibonacci(k-3). - Ehren Metcalfe, Apr 15 2019

Product_{n>=1} (1 - (-1)^n/a(n)) = sin(3*Pi/10) (A019863). - Amiram Eldar, Nov 28 2024