A007660 a(n) = a(n-1)*a(n-2) + 1 with a(0) = a(1) = 0.

0, 0, 1, 1, 2, 3, 7, 22, 155, 3411, 528706, 1803416167, 953476947989903, 1719515742866809222961802, 1639518622529236077952144318816050685207, 2819178082162327154499022366029959843954512194276761760087463015

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

If we omit the first three terms of the sequence, a(n)/a(n-1) can be expressed as the continued fraction [a(n-2); a(n-1)]. - Eric Angelini, Feb 10 2005

This may be regarded as a multiplicative dual of the Fibonacci sequence A000045. Write Fibonacci's formula as F(0)=0, F(1)=1; F(n)=[F(n-1)+F(n-2)]*1 with n>1. Swap '+' and '*' and we have the present sequence! - B. Joshipura (bhushit(AT)yahoo.com), Aug 29 2007

a(n+1) divides a(2n+1), a(3n+1), a(4n+1), etc., this is because modulo a(n+1): a(1)=a(n+1)=0 and a(2)=a(n+2)=1 so the sequence repeats modulo a(n+1) with period n. - Isaac Kaufmann, Sep 04 2020

Examples

			b(10) / b(5) = 1803416167 / 7 = 257630881. - _Michael Somos_, Dec 29 2012
		

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..19 (shortened by _N. J. A. Sloane_, Jan 13 2019)
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
• Bhushit Joshipura, 1, 1, 2, 3, 7, ... Multiplication dual of Fibonacci?, posting in newsgroup sci.math, Jul 28 2007.
• S. Kak, The Golden Mean and the Physics of Aesthetics, arXiv:physics/0411195 [physics.hist-ph], 2004.

Crossrefs

Cf. A250309, A253853, A258113.

Programs

Haskell

a007660 n = a007660_list !! n
a007660_list = 0 : 0 : map (+ 1)
                       (zipWith (*) a007660_list $ tail a007660_list)
-- Reinhard Zumkeller, Jan 17 2015

Magma

I:=[0,0]; [n le 2 select I[n] else Self(n-1)*Self(n-2)+1: n in [1..20]]; // Vincenzo Librandi, Nov 14 2011

Mathematica

a[0] = a[1] = 0; a[n_] := a[n - 1]*a[n - 2] + 1; Table[ a[n], {n, 0, 15} ]
RecurrenceTable[{a[0]==a[1]==0,a[n]==a[n-1]a[n-2]+1},a,{n,20}] (* Harvey P. Dale, Nov 12 2011 *)

Maxima

a(n) := if (n=0 or n=1) then 0 else a(n-1)*a(n-2)+1 $
makelist(a(n),n,0,18); /* Emanuele Munarini, Mar 24 2017 */

Formula

a(n) is asymptotic to c^(phi^n) where phi = (1 + sqrt(5))/2 and c = A258113 = 1.1130579759029319... - Benoit Cloitre, Sep 26 2003

b(n) = a(n+1) is a divisibility sequence. - Michael Somos, Dec 29 2012