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A000301 a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n). 37
1, 2, 2, 4, 8, 32, 256, 8192, 2097152, 17179869184, 36028797018963968, 618970019642690137449562112, 22300745198530623141535718272648361505980416, 13803492693581127574869511724554050904902217944340773110325048447598592 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Continued fraction expansion of s = 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

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

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

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 (2011).

Index to divisibility sequences

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

MAPLE

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

MATHEMATICA

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

PROG

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

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

(Haskell)

a000301 = a000079 . a000045

a000301_list = 1 : scanl (*) 2 a000301_list

-- Reinhard Zumkeller, Mar 20 2013

CROSSREFS

Cf. A000045, A010098, A010099, A010100.

Cf. A000079.

Column k = 2 of A244003.

Sequence in context: A070323 A109213 A109214 * A124439 A082836 A201376

Adjacent sequences:  A000298 A000299 A000300 * A000302 A000303 A000304

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Mar 15 1996

EXTENSIONS

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

STATUS

approved

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Last modified May 23 23:53 EDT 2017. Contains 286937 sequences.