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A001611 Fibonacci(n) + 1.
(Formerly M0288 N0103)
54
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 (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) [From Roland Schroeder (florola(AT)gmx.de), Aug 05 2010]

REFERENCES

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

Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 5.1.

D. Jarden, Recurring Sequences. Riveon Lematematika, Jerusalem, 1966.

N. S. Mendelsohn, Permutations with restricted displacement, Canad. Math. Bull., 4 (1961), 29-38.

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 402

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index to sequences with 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) [From Roland Schroeder (florola(AT)gmx.de), Aug 05 2010]

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.

Sequence in context: A212264 A174650 A107293 * 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 September 17 11:28 EDT 2014. Contains 246841 sequences.