OFFSET
0,1
COMMENTS
Let b = sqrt(5)/5. We have the alternating series identity (10 - 4*sqrt(5))/5 = b/2 - b^2/(2*8) + b^3/(2*8*144) - b^4/(2*8*144*46368) + ..., so this sequence is a generalized Pierce expansion of (10 - 4*sqrt(5))/5 to the base b as defined in A058635. - Peter Bala, Nov 04 2013
REFERENCES
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete mathematics, second edition, Addison Wesley, 1994, p. 557, ex. 6.61.
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..10
V. E. Hoggatt, Jr. and Marjorie Bicknell, A Reciprocal Series of Fibonacci Numbers with Subscripts 2^n k, The Fibonacci Quarterly, Vol. 14, No. 5 (1976), p. 453-455.
FORMULA
Sum_{n>=0} 1/a(n) = 5/4 - 1/phi = 0.6319660112... since Sum_{k=0..n} 1/a(k) = 5/4 - F(3*2^n-1)/F(3*2^n).
a(n) = (1/sqrt(5))*( (2 + sqrt(5))^2^n - 1/(2 + sqrt(5))^2^n ) for n >= 1. - Peter Bala, Nov 04 2013
MATHEMATICA
Table[Fibonacci[3*2^n], {n, 0, 7}] (* Amiram Eldar, Jan 29 2022 *)
PROG
(Magma) [Fibonacci(3*2^n) : n in [0..7]]; // Wesley Ivan Hurt, Apr 05 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Jan 29 2003
STATUS
approved