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%I M0853 #62 Sep 08 2022 08:44:35
%S 0,0,1,1,2,3,7,22,155,3411,528706,1803416167,953476947989903,
%T 1719515742866809222961802,1639518622529236077952144318816050685207,
%U 2819178082162327154499022366029959843954512194276761760087463015
%N a(n) = a(n-1)*a(n-2) + 1 with a(0) = a(1) = 0.
%C 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
%C 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
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A007660/b007660.txt">Table of n, a(n) for n = 0..19</a> (shortened by _N. J. A. Sloane_, Jan 13 2019)
%H A. V. Aho and N. J. A. Sloane, <a href="https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fib. Quart., 11 (1973), 429-437.
%H Bhushit Joshipura, <a href="https://groups.google.com/g/sci.math/c/QkTxa20XQ3Q/m/XkPmSVtNVFYJ">1, 1, 2, 3, 7, ... Multiplication dual of Fibonacci?</a>, posting in newsgroup sci.math, Jul 28 2007.
%H S. Kak, <a href="https://arxiv.org/abs/physics/0411195">The Golden Mean and the Physics of Aesthetics</a>, arXiv:physics/0411195 [physics.hist-ph], 2004.
%F a(n) is asymptotic to c^(phi^n) where phi = (1 + sqrt(5))/2 and c = A258113 = 1.1130579759029319... - _Benoit Cloitre_, Sep 26 2003
%F b(n) = a(n+1) is a divisibility sequence. - _Michael Somos_, Dec 29 2012
%e b(10) / b(5) = 1803416167 / 7 = 257630881. - _Michael Somos_, Dec 29 2012
%t a[0] = a[1] = 0; a[n_] := a[n - 1]*a[n - 2] + 1; Table[ a[n], {n, 0, 15} ]
%t RecurrenceTable[{a[0]==a[1]==0,a[n]==a[n-1]a[n-2]+1},a,{n,20}] (* _Harvey P. Dale_, Nov 12 2011 *)
%o (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
%o (Haskell)
%o a007660 n = a007660_list !! n
%o a007660_list = 0 : 0 : map (+ 1)
%o (zipWith (*) a007660_list $ tail a007660_list)
%o -- _Reinhard Zumkeller_, Jan 17 2015
%o (Maxima) a(n) := if (n=0 or n=1) then 0 else a(n-1)*a(n-2)+1 $
%o makelist(a(n),n,0,18); /* _Emanuele Munarini_, Mar 24 2017 */
%Y Cf. A250309, A253853, A258113.
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_, _Robert G. Wilson v_
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