|
| |
|
|
A001611
|
|
Fibonacci(n) + 1.
(Formerly M0288 N0103)
|
|
28
| |
|
|
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; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| a(n) = Fibonacci(n) + 1.
a(0) = 1, a(1) = 2 then the largest number such that a triangle is constructible with three successive terms as sides. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 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
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 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); [S. 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 (* From 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
|
|
|
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]
Sequence in context: A157876 A174650 A107293 * A039829 A143588 A032245
Adjacent sequences: A001608 A001609 A001610 * A001612 A001613 A001614
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
