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

1,3

Comments

Consider the mapping f(a/b) = (a+b) / (a*b). 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

From Ali Pourzand, Nov 19 2025: (Start)

Geometric interpretation: start with a 1 X 1 square. At each step, form a new rectangle by attaching to the current rectangle a(n) copies of itself to the right, and in the next row, b(n) copies of itself rotated by 90 degrees. If the current rectangle has side lengths (a,b), then the next rectangle has dimensions (a',b') = (a*b,a+b). Thus the pair (a(n),b(n)) gives the successive rectangle dimensions obtained by this iterative "self-attachment" process.

(1,1) (1,2) (2,3) (5,6)

_ _ _ _ _ _ _ _ _ _ _

|_| |_ _| |_ _| | | |

| | | |_ _ _|_ _ _|

|_|_| | | | |

| | | |

|_ _|_ _|_ _|

The sequence a(n) records the lengths of one side of these rectangles, with (a(n),b(n)) = (1,1),(1,2),(2,3),(6,5),(30,11), ... (End)

Links

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

Formula

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

Limit_{n -> oo} 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

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 *)

Prog

(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
(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
(SageMath)
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
(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

Crossrefs

The b(n) sequence is A003686.

See A094303 for another version.

Cf. A001697, A070231, A070233, A070234.

Cf. A001622 (golden ratio).

Sequence in context: A329558 A038696 A333373 * A242996 A127815 A054934

Adjacent sequences: A064844 A064845 A064846 * A064848 A064849 A064850

Keyword

nonn,easy

Author

Leroy Quet, Oct 31 2001

Status

approved