A000301 a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n).

1, 2, 2, 4, 8, 32, 256, 8192, 2097152, 17179869184, 36028797018963968, 618970019642690137449562112, 22300745198530623141535718272648361505980416, 13803492693581127574869511724554050904902217944340773110325048447598592

Offset: 0

Author

N. J. A. Sloane, Mar 15 1996

Keywords

• nonn
• easy

Comments

Continued fraction expansion of s = A073115 = 1.709803442861291... = Sum_{k >= 0} (1/2^floor(k * phi)) where phi is the golden ratio (1 + sqrt(5))/2. - Benoit Cloitre, Aug 19 2002

The continued fraction expansion of the above constant s is [1; 1, 2, 2, 4, ...], that of the rabbit constant r = s-1 = A014565 is [0; 1, 2, 2, 4, ...]. - M. F. Hasler, Nov 10 2018

References

• Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 913.

Links

• T. D. Noe, Table of n, a(n) for n = 0..18
• Manosij Ghosh Dastidar and Michael Wallner, Bijections between Variants of Dyck Paths and Integer Compositions, arXiv:2406.16404 [math.CO], 2024. See p. 1.
• J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc., 63 (1977), 29-32.
• Samuele Giraudo, Intervals of balanced binary trees in the Tamari lattice, arXiv preprint arXiv:1107.3472 [math.CO], 2011-2012, and Theor Comput Sci 420 (2012) 1-27.
• Bertrand Teguia Tabuguia, Computing with D-Algebraic Sequences, arXiv:2412.20630 [math.AG], 2024. See p. 9.
• Index to divisibility sequences

Crossrefs

Cf. A000045, A000079, A000304, A010098, A010099, A010100, A073115, A124091.

Column k = 2 of A244003.

Programs

Haskell

a000301 = a000079 . a000045
a000301_list = 1 : scanl (*) 2 a000301_list
-- Reinhard Zumkeller, Mar 20 2013

Magma

[2^Fibonacci(n): n in [0..20]]; // Vincenzo Librandi, Apr 18 2011

Maple

A000301 := proc(n) option remember;
             if n < 2 then 1+n
           else A000301(n-1)*A000301(n-2)
             fi
           end:
seq(A000301(n), n=0..15);

Mathematica

2^Fibonacci[Range[0, 14]] (* Alonso del Arte, Jul 28 2016 *)

PARI

a(n)=1<Charles R Greathouse IV, Jan 12 2012

SageMath

[2^fibonacci(n) for n in range(15)] # G. C. Greubel, Jul 29 2024

Formula

a(n) ~ k^phi^n with k = 2^(1/sqrt(5)) = 1.3634044... and phi the golden ratio. - Charles R Greathouse IV, Jan 12 2012

a(n) = A000304(n+3) / A010098(n+1). - Reinhard Zumkeller, Jul 06 2014

Sum_{n>=0} 1/a(n) = A124091. - Amiram Eldar, Oct 27 2020

Limit_{n->oo} a(n)/a(n-1)^phi = 1. - Peter Woodward, Nov 24 2023

Extensions

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 18 2011