A007570 a(n) = F(F(n)), where F is a Fibonacci number. (Formerly M1537)

0, 1, 1, 1, 2, 5, 21, 233, 10946, 5702887, 139583862445, 1779979416004714189, 555565404224292694404015791808, 2211236406303914545699412969744873993387956988653, 2746979206949941983182302875628764119171817307595766156998135811615145905740557

Offset

0,5

Comments

a(20) is approximately 2.830748520089124 * 10^1413, much too large to include even in the b-file. - Alonso del Arte, Apr 30 2020

Let M(0) denote the 2 X 2 identity matrix, and let M(1) = [[0, 1], [1, 1]]. Let M(n) = M(n-2) * M(n-1). Then a(n) is equal to both the (1, 2)-entry and the (2, 1)-entry of M(n). - John M. Campbell, Jul 02 2021

This is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Dec 06 2022

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..19 (terms n = 0..17 from T. D. Noe)

Alonso del Arte, Table of n, a(n) for n = 0 .. 24, with digits grouped in hundreds

John M. Campbell, A Matrix-Based Recursion Relation for F_{F_n}, Fib. Quart., Vol. 60, No. 3 (2022), pp. 256-261.

Bakir Farhi, Summation of certain infinite Fibonacci related series, arXiv:1512.09033 [math.NT], 2015. See p. 6, eq. 2.9.

George Ledin, Jr., Problem H-147, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 6, No. 6 (1968), p. 352; Converging Fractions, Solution to Problem H-147 by David Zeitlin, ibid., Vol. 8, No. 4 (1970), pp. 387-389.

Edward A. Parberry, Two recursion relations for F(F(n)), Fib. Quart., Vol. 15, No. 2 (1977), p. 122 and p. 139.

Martin Stein, Algebraic independence results for reciprocal sums of Fibonacci and Lucas numbers, Dissertation, Hannover: Gottfried Wilhelm Leibniz Universität Hannover, 2012.

Chris Street, A Recurrence for the Sequence {F(F(n)),n=>0}.

Formula

a(n+1)/a(n) ~ phi^(F(n-1)), with phi = (1 + sqrt(5))/2 = A001622. - Carmine Suriano, Jan 24 2011

Sum_{n>=1} 1/a(n) = 3.7520024260... is transcendental (Stein, 2012). - Amiram Eldar, Oct 30 2020

Sum_{n>=1} (-1)^(F(n)+1)*a(n-1)/(a(n)*a(n+1)) = 1/phi (A094214) (Farhi, 2015). - Amiram Eldar, Apr 07 2021

Limit_{n->oo} a(n+1)/a(n)^phi = 5^((phi-1)/2) = 1.6443475285..., where phi is the golden ratio (A001622) (Ledin, 1968) - Amiram Eldar, Feb 02 2022

Maple

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= n-> F(F(n)):
seq(a(n), n=0..14);  # Alois P. Heinz, Oct 09 2015

Mathematica

F[0] = 0; F[1] = 1; F[n_] := F[n] = F[n - 1] + F[n - 2]; Table[F[F[n]], {n, 0, 14}]
Fibonacci[Fibonacci[Range[0, 20]]] (* Harvey P. Dale, May 05 2012 *)

Prog

(Sage) [fibonacci(fibonacci(n)) for n in range(0, 14)] # Zerinvary Lajos, Nov 30 2009
(PARI) a(n)=fibonacci(fibonacci(n)) \\ Charles R Greathouse IV, Feb 03 2014
(Scala) val fibo: LazyList[BigInt] = (0: BigInt) #:: (1: BigInt) #:: fibo.zip(fibo.tail).map { n => n._1 + n._2 }
val fiboLimited: LazyList[Int] = 0 #:: 1 #:: fiboLimited.zip(fiboLimited.tail).map { n => n._1 + n._2 } // Limited to 32-bit integers because that's the type for LazyList apply()
(0 to 19).map(n => fibo(fiboLimited(n))) // Alonso del Arte, Apr 30 2020
(Python)
from sympy import fibonacci
def a(n): return fibonacci(fibonacci(n))
print([a(n) for n in range(15)]) # Michael S. Branicky, Feb 02 2022

Crossrefs

Cf. A000045, A001622, A005371, A058051, A094214.

Sequence in context: A162437 A357871 A216756 * A173313 A210575 A174143

Adjacent sequences: A007567 A007568 A007569 * A007571 A007572 A007573

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Robert G. Wilson v

Extensions

One more term from Harvey P. Dale, May 05 2012

Status

approved