A121821 -id:A121821 - cp's OEIS frontend

4, 2, 8, 16, 128, 2048, 262144, 536870912, 140737488355328, 75557863725914323419136, 10633823966279326983230456482242756608, 803469022129495137770981046170581301261101496891396417650688

Offset: 0

Peter Bala, Oct 31 2013

Compare with A000301(n) = 2^Fibonacci(n).
The sequence a(n) for n >= 1 gives the sequence of partial quotients (other than the first) in the continued fraction expansion of the transcendental real constant c := sum {n >= 1} 1/2^floor(n*(5 + sqrt(5))/2) = 0.13385 44229 67609 80592 ... = 1/(7 + 1/(2 + 1/(8 + 1/(16 + 1/(128 + 1/(2048 + ...)))))). See Adams Davison 1977. Cf. A014565.
The constant c has various series representations including
c = 1 - sum {n >= 1} 1/2^floor(n*(5 - sqrt(5))/2),
c = sum {n >= 1} floor(n*(5 - sqrt(5))/10)/2^n,
c = 3 - sum {n >= 1} 1/2^floor(n*(15 - sqrt(5))/22) and
c = sum {n >= 1} 1/2^floor(n*(15 + sqrt(5))/22) - 2.

William 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.
J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc. 63 (1977), pp. 29-32.

Cf. A000032, A000301, A014565, A121821.

[2^(Lucas(n)): n in [0..10]]; // G. C. Greubel, Dec 22 2017

a := proc(n) option remember; if n = 0 then 4 elif n = 1 then 2 else a(n-1)*a(n-2); fi; end; seq(a(n), n = 0..10);

2^LucasL[Range[0,15]] (* Harvey P. Dale, Jul 21 2015 *)

for(n=0,10, print1(2^(fibonacci(n+1) + fibonacci(n-1)), ", ")) \\ G. C. Greubel, Dec 22 2017

a(n) = 2^Lucas(n) = 2^A000032(n).
Recurrence: a(n) = a(n-1)*a(n-2) with a(0) = 4, a(1) = 2.
Sum_{n>=1} 1/a(n) = A121821. - Amiram Eldar, Oct 27 2020

cp's OEIS Frontend

A230900 a(n) = 2^Lucas(n).

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