A007660 a(n) = a(n-1)*a(n-2) + 1 with a(0) = a(1) = 0. (Formerly M0853)

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

Offset

0,5

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

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, Fib. Quart., 11 (1973), 429-437.

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.

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A250309, A253853, A258113.

Sequence in context: A151908 A072214 A233535 * A158055 A156615 A158054

Adjacent sequences: A007657 A007658 A007659 * A007661 A007662 A007663

Keyword

nonn,easy

Author

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

Status

approved