A007570 - OEIS

%I M1537 #125 Oct 29 2024 12:06:19

%S 0,1,1,1,2,5,21,233,10946,5702887,139583862445,1779979416004714189,

%T 555565404224292694404015791808,

%U 2211236406303914545699412969744873993387956988653,2746979206949941983182302875628764119171817307595766156998135811615145905740557

%N a(n) = F(F(n)), where F is a Fibonacci number.

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

%C 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

%C 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

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

%H Alois P. Heinz, <a href="/A007570/b007570.txt">Table of n, a(n) for n = 0..19</a> (terms n = 0..17 from T. D. Noe)

%H Alonso del Arte, <a href="/A007570/a007570.txt">Table of n, a(n) for n = 0 .. 24, with digits grouped in hundreds</a>

%H John M. Campbell, <a href="/A007570/a007570.pdf">A Matrix-Based Recursion Relation for F_{F_n}</a>, Fib. Quart., Vol. 60, No. 3 (2022), pp. 256-261.

%H Bakir Farhi, <a href="https://arxiv.org/abs/1512.09033">Summation of certain infinite Fibonacci related series</a>, arXiv:1512.09033 [math.NT], 2015. See p. 6, eq. 2.9.

%H George Ledin, Jr., <a href="https://fq.math.ca/Scanned/6-6/advanced6-6.pdf">Problem H-147</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 6, No. 6 (1968), p. 352; <a href="https://www.fq.math.ca/Scanned/8-4/advanced8-4.pdf">Converging Fractions</a>, Solution to Problem H-147 by David Zeitlin, ibid., Vol. 8, No. 4 (1970), pp. 387-389.

%H Edward A. Parberry, Two recursion relations for F(F(n)), Fib. Quart., Vol. 15, No. 2 (1977), <a href="http://www.fq.math.ca/Scanned/15-2/parberry-a.pdf">p. 122</a> and <a href="http://www.fq.math.ca/Scanned/15-2/parberry-b.pdf">p. 139</a>.

%H Martin Stein, <a href="https://doi.org/10.15488/7983">Algebraic independence results for reciprocal sums of Fibonacci and Lucas numbers</a>, Dissertation, Hannover: Gottfried Wilhelm Leibniz Universität Hannover, 2012.

%H Chris Street, <a href="http://www.codehappy.net/fibo.pdf">A Recurrence for the Sequence {F(F(n)), n>=0}</a>.

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

%F Sum_{n>=1} 1/a(n) = 3.7520024260... is transcendental (Stein, 2012). - _Amiram Eldar_, Oct 30 2020

%F 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

%F 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

%p F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:

%p a:= n-> F(F(n)):

%p seq(a(n), n=0..14);  # _Alois P. Heinz_, Oct 09 2015

%t F[0] = 0; F[1] = 1; F[n_] := F[n] = F[n - 1] + F[n - 2]; Table[F[F[n]], {n, 0, 14}]

%t Fibonacci[Fibonacci[Range[0, 20]]] (* _Harvey P. Dale_, May 05 2012 *)

%o (Sage) [fibonacci(fibonacci(n)) for n in range(0, 14)] # _Zerinvary Lajos_, Nov 30 2009

%o (PARI) a(n)=fibonacci(fibonacci(n)) \\ _Charles R Greathouse IV_, Feb 03 2014

%o (Scala) val fibo: LazyList[BigInt] = (0: BigInt) #:: (1: BigInt) #:: fibo.zip(fibo.tail).map { n => n._1 + n._2 }

%o 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()

%o (0 to 19).map(n => fibo(fiboLimited(n))) // _Alonso del Arte_, Apr 30 2020

%o (Python)

%o from sympy import fibonacci

%o def a(n): return fibonacci(fibonacci(n))

%o print([a(n) for n in range(15)]) # _Michael S. Branicky_, Feb 02 2022

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

%K nonn,nice,easy

%O 0,5

%A _N. J. A. Sloane_, _Robert G. Wilson v_

%E One more term from _Harvey P. Dale_, May 05 2012