A010100 - OEIS

A010100

a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=10.

1, 10, 10, 100, 1000, 100000, 100000000, 10000000000000, 1000000000000000000000, 10000000000000000000000000000000000, 10000000000000000000000000000000000000000000000000000000

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OFFSET

0,2

COMMENTS

From Peter Bala, Nov 11 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 := 9*sum {n = 1..inf} 1/10^floor(n*phi) (= 81*sum {n = 1..inf} floor(n/phi)/10^n) = 0.90990 90990 99090 99090 ... = 1/(1 + 1/(10 + 1/(10 + 1/(100 + 1/(1000 + 1/(100000 + 1/(100000000 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565 and A005614.

Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/10^(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

Alois P. Heinz, 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) = 10^Fibonacci(n).

MAPLE

a:= n-> 10^(<<1|1>, <1|0>>^n)[1, 2]:

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

MATHEMATICA

10^Fibonacci[Range[0, 10]] (* Harvey P. Dale, Feb 12 2023 *)

PROG

(PARI) a(n) = 10^fibonacci(n); \\ Michel Marcus, Oct 25 2017

CROSSREFS