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A010098 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=3. 13
1, 3, 3, 9, 27, 243, 6561, 1594323, 10460353203, 16677181699666569, 174449211009120179071170507, 2909321189362570808630465826492242446680483, 507528786056415600719754159741696356908742250191663887263627442114881 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Peter Bala, Nov 01 2013: (Start)

Let phi = (1/2)*(1 + sqrt(5)) denote the golden ratio A001622. This sequence gives the simple continued fraction expansion of the constant c := 2*Sum_{n>=1} 1/3^floor(n*phi) (= 4*Sum_{n>=1} floor(n/phi)/3^n) = 0.768597560593155198508 ... = 1/(1 + 1/(3 + 1/(3 + 1/(9 + 1/(27 + 1/(243 + 1/(6561 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.

Furthermore, for k = 0,1,2,... if we put X(k) = sum {n >= 1} 1/3^(n*Fibonacci(k) + Fibonacci(k+1)*floor(n*phi)) then the real number X(k+1)/X(k) has the simple continued fraction expansion [0; a(k+1), a(k+2), a(k+3), ...] (apply Bowman 1988, Corollary 1). (End)

a(n+1) = A000304(n+3) / A000301(n). - Reinhard Zumkeller, Jul 06 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..17

W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.

P. G. Anderson, T. C. Brown, P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.

D. Bowman, A new generalization of Davison's theorem, Fib. Quart. Volume 26 (1988), 40-45

FORMULA

a(n) = 3^Fibonacci(n).

MAPLE

a[ -1]:=1:a[0]:=3: a[1]:=3: for n from 2 to 13 do a[n]:=a[n-1]*a[n-2] od: seq(a[n], n=-1..10); # Zerinvary Lajos, Mar 19 2009

MATHEMATICA

3^Fibonacci[Range[0, 13]] (* or *) t={1, 3}; Do[AppendTo[t, t[[-1]]*t[[-2]]], {n, 12}]; t (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)

PROG

(Haskell)a010098 n = a010098_list !! n

a010098_list = 1 : 3 : zipWith (*) a010098_list (tail a010098_list)

-- Reinhard Zumkeller, Jul 06 2014

CROSSREFS

Cf. A000045, A000301, A010099, A010100, A014565, A214706, A214887, A215270, A215271, A215272.

Column k=3 of A244003.

Sequence in context: A268092 A229024 A117976 * A029857 A327712 A257611

Adjacent sequences:  A010095 A010096 A010097 * A010099 A010100 A010101

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 25 08:35 EDT 2020. Contains 334585 sequences. (Running on oeis4.)