A079613 a(n) = F(3*2^n) where F(k) denotes the k-th Fibonacci number.

2, 8, 144, 46368, 4807526976, 51680708854858323072, 5972304273877744135569338397692020533504, 79757008057644623350300078764807923712509139103039448418553259155159833079730688

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) = A081976(n+1)*A081976(n+2). - Amarnath Murthy, Apr 03 2003

a(n) = (1/sqrt(5))*( (2 + sqrt(5))^2^n - 1/(2 + sqrt(5))^2^n ) for n >= 1. - Peter Bala, Nov 04 2013

a(n) = A000045(A007283(n)). - Amiram Eldar, Jan 29 2022

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

Cf. A000045, A001622 (phi), A007283, A058635, A081976, A083523.

Sequence in context: A166356 A009817 A124105 * A091299 A362729 A307326

Adjacent sequences: A079610 A079611 A079612 * A079614 A079615 A079616

Keyword

nonn

Author

Benoit Cloitre, Jan 29 2003

Status

approved