A064847 Sequence a(n) such that there is a sequence b(n) with a(1) = b(1) = 1, a(n+1) = a(n) * b(n) and b(n+1) = a(n) + b(n) for n >= 1.

1, 1, 2, 6, 30, 330, 13530, 5019630, 69777876630, 351229105131280530, 24509789089304573335878465330, 8608552999157278550998626549630446732052243030

Offset: 1

Author

Leroy Quet, Oct 31 2001

Keywords

• nonn
• easy

Comments

Consider the mapping f(a/b) = (a + b)/(ab). Taking a = 1 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 2/1, 3/2, 5/6, 11/30, ... The current sequence contains the denominators. - Amarnath Murthy, Mar 24 2003

Links

• Harry J. Smith, Table of n, a(n) for n=1..18

Crossrefs

The b(n) sequence is A003686.

See A094303 for another version.

Cf. A001697, A070231, A070233, A070234.

Cf. A001622 (golden ratio).

Programs

Haskell

a064847 n = a064847_list !! (n-1)
a064847_list = 1 : f [1,1] where
   f xs'@(x:xs) = y : f (y : xs') where y = x * sum xs
-- Reinhard Zumkeller, Apr 29 2013

Magma

[n le 2 select 1 else Self(n-1)*(Self(n-1)/Self(n-2) + Self(n-2)): n in [1..14]]; // Vincenzo Librandi, Dec 17 2015

Maple

f:= proc(n) option remember; procname(n-1)*(procname(n-1)/procname(n-2) + procname(n-2)) end proc:
f(1):= 1: f(2):= 1:
map(f, [$1..16]); # Robert Israel, Jul 18 2016

Mathematica

RecurrenceTable[{a[n]==a[n-1]*(a[n-1]/a[n-2] + a[n-2]), a[0]==1, a[1]==1},a,{n,0,15}] (* Vaclav Kotesovec, May 21 2015 *)
Im[NestList[Re@#+(1+I Re@#)Im@#&, 1+I, 15]] (* Vladimir Reshetnikov, Jul 18 2016 *)

PARI

{ for (n=1, 18, if (n>2, a=a1*(a1/a2 + a2); a2=a1; a1=a, a=a1=a2=1); write("b064847.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 28 2009

Sage

def A064847():
    x, y = 1, 2
    yield x
    while True:
        yield x
        x, y = x * y, x + y
a = A064847()
[next(a) for i in range(12)]  # Peter Luschny, Dec 17 2015

Formula

a(n+2) = a(n+1)*(a(n+1)/a(n) + a(n)) for n >= 1 with a(1) = a(2) = 1.

Lim_{n -> infinity} a(n)/A003686(n)^phi = 1, where phi = (1 + sqrt(5))/2 is the golden ratio. - Benoit Cloitre, May 08 2002

Denominator of b(n), where b(n) = 1/numerator(b(n-1)) + 1/denominator(b(n-1)) for n >= 2 with b(1) = 1. Cf. A003686. - Vladeta Jovovic, Aug 15 2002

a(n) ~ c^(phi^n), where c = 1.70146471458872503754529013562504670973656402413202907200954401051557047249... and phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015