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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.
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%I #56 Sep 08 2022 08:45:04

%S 1,1,2,6,30,330,13530,5019630,69777876630,351229105131280530,

%T 24509789089304573335878465330,

%U 8608552999157278550998626549630446732052243030

%N 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.

%C 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

%H Harry J. Smith, <a href="/A064847/b064847.txt">Table of n, a(n) for n=1..18</a>

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

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

%F 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

%F 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

%p f:= proc(n) option remember; procname(n-1)*(procname(n-1)/procname(n-2) + procname(n-2)) end proc:

%p f(1):= 1: f(2):= 1:

%p map(f, [$1..16]); # _Robert Israel_, Jul 18 2016

%t 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 *)

%t Im[NestList[Re@#+(1+I Re@#)Im@#&, 1+I, 15]] (* _Vladimir Reshetnikov_, Jul 18 2016 *)

%o (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

%o (Haskell)

%o a064847 n = a064847_list !! (n-1)

%o a064847_list = 1 : f [1,1] where

%o f xs'@(x:xs) = y : f (y : xs') where y = x * sum xs

%o -- _Reinhard Zumkeller_, Apr 29 2013

%o (Sage)

%o def A064847():

%o x, y = 1, 2

%o yield x

%o while True:

%o yield x

%o x, y = x * y, x + y

%o a = A064847()

%o [next(a) for i in range(12)] # _Peter Luschny_, Dec 17 2015

%o (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

%Y The b(n) sequence is A003686.

%Y See A094303 for another version.

%Y Cf. A001697, A070231, A070233, A070234.

%Y Cf. A001622 (golden ratio).

%K nonn,easy

%O 1,3

%A _Leroy Quet_, Oct 31 2001