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A001611 a(n) = Fibonacci(n) + 1.
(Formerly M0288 N0103)
72
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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

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 *)

PROG

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

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

(Haskell)

a001611 = (+ 1) . a000045

a001611_list = 1 : 2 : map (subtract 1)

                       (zipWith (+) a001611_list $ tail a001611_list)

-- Reinhard Zumkeller, Jul 30 2013

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, A160536, A212272, A242876.

Sequence in context: A329693 A329976 A329703 * A214448 A039829 A143588

Adjacent sequences:  A001608 A001609 A001610 * A001612 A001613 A001614

KEYWORD

nonn,easy,hear

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)