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A214706 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=5. 9
1, 5, 5, 25, 125, 3125, 390625, 1220703125, 476837158203125, 582076609134674072265625, 277555756156289135105907917022705078125, 161558713389263217748322010169914619837072677910327911376953125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(17) has 1117 digits.

From Peter Bala, Nov 01 2013: (Start)

Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 4*sum {n = 1..inf} 1/5^floor(n*phi) (= 16*sum {n = 1..inf} floor(n/phi)/5^n) = 0.83866 83869 91037 14262 ... = 1/(1 + 1/(5 + 1/(5 + 1/(25 + 1/(125 + 1/(3125 + 1/(390625 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.

Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/5^(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)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..16

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) = 5^Fibonacci(n).

MAPLE

a:= n-> 5^(<<1|1>, <1|0>>^n)[1, 2]:

seq(a(n), n=0..12);  # Alois P. Heinz, Jun 17 2014

MATHEMATICA

5^Fibonacci[Range[0, 11]]

nxt[{a_, b_}]:={b, a*b}; NestList[nxt, {1, 5}, 12][[All, 1]] (* Harvey P. Dale, Oct 14 2018 *)

PROG

(Magma) [5^Fibonacci(n): n in [0..13]];

CROSSREFS

Cf. A000045, A000301, A010098, A010099, A010100, A214887, A014565, A214706, A215270, A215271, A215272.

Column k=5 of A244003.

Sequence in context: A257624 A176160 A222281 * A203191 A165423 A220078

Adjacent sequences:  A214703 A214704 A214705 * A214707 A214708 A214709

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Aug 01 2012

STATUS

approved

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Last modified September 26 22:02 EDT 2022. Contains 357051 sequences. (Running on oeis4.)