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A215272 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=9.

%I A215272 #43 Jan 05 2025 19:51:39
%S A215272 1,9,9,81,729,59049,43046721,2541865828329,109418989131512359209,
%T A215272 278128389443693511257285776231761,
%U A215272 30432527221704537086371993251530170531786747066637049
%N A215272 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=9.
%C A215272 From _Peter Bala_, Nov 01 2013: (Start)
%C A215272 Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 8*sum {n = 1..inf} 1/9^floor(n*phi) (= 64*sum {n = 1..inf} floor(n/phi)/9^n) = 0.90109 74122 99938 29901 ... = 1/(1 + 1/(9 + 1/(9 + 1/(81 + 1/(729 + 1/(59049 + 1/(43046721 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
%C A215272 Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/9^(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)
%H A215272 Bruno Berselli, <a href="/A215272/b215272.txt">Table of n, a(n) for n = 0..15</a>
%H A215272 W. W. Adams and J. L. Davison, <a href="http://www.jstor.org/stable/2041889">A remarkable class of continued fractions</a>, Proc. Amer. Math. Soc. 65 (1977), 194-198.
%H A215272 P. G. Anderson, T. C. Brown, P. J.-S. Shiue, <a href="http://people.math.sfu.ca/~vjungic/tbrown/tom-28.pdf">A simple proof of a remarkable continued fraction identity</a>, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
%H A215272 D. Bowman, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/26-1/bowman.pdf">A new generalization of Davison's theorem</a>, Fib. Quart. Volume 26 (1988), 40-45
%F A215272 a(n) = 9^Fibonacci(n).
%p A215272 a:= n-> 9^(<<1|1>, <1|0>>^n)[1, 2]:
%p A215272 seq(a(n), n=0..12);  # _Alois P. Heinz_, Jun 17 2014
%t A215272 RecurrenceTable[{a[0] == 1, a[1] == 9, a[n] == a[n - 1] a[n - 2]}, a[n], {n, 0, 15}]
%o A215272 (Magma) [9^Fibonacci(n): n in [0..10]];
%o A215272 (PARI) a(n) = 9^fibonacci(n); \\ _Jinyuan Wang_, Apr 06 2019
%Y A215272 Cf. A000045, A000301, A010098-A010100, A214706, A214887, A215270, A215271, A014565.
%Y A215272 Column k=9 of A244003.
%K A215272 nonn,easy
%O A215272 0,2
%A A215272 _Bruno Berselli_, Aug 07 2012