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A001056 a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.
(Formerly M2378 N0944)
3

%I M2378 N0944

%S 1,3,4,13,53,690,36571,25233991,922832284862,23286741570717144243,

%T 21489756930695820973683319349467,

%U 500426416062641238759467086706254193219790764168482,10754042042885415070816603338436200915110904821126871858491675028294447933424899095

%N a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.

%D Archimedeans Problems Drive, Eureka, 19 (1957), 13.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H T. D. Noe, <a href="/A001056/b001056.txt">Table of n, a(n) for n = 0..17</a>

%H A. V. Aho and N. J. A. Sloane, <a href="http://neilsloane.com/doc/doubly.html">Some doubly exponential sequences</a>, Fib. Quart., 11 (1973), 429-437.

%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>

%F a(n) ~ c^(phi^n), where c = A258112 = 1.7978784900091604813559508837..., phi = (1+sqrt(5))/2 = A001622. - _Vaclav Kotesovec_, Dec 17 2014

%p a:= proc (n) option remember;

%p if n=0 then 1

%p elif n=1 then 3

%p else a(n-1)*a(n-2) + 1

%p end if

%p end proc;

%p seq(a(n), n = 0..13); # _G. C. Greubel_, Sep 19 2019

%t RecurrenceTable[{a[0]==1,a[1]==3,a[n]==a[n-1]*a[n-2]+1},a,{n,0,14}] (* _Harvey P. Dale_, Jul 17 2011 *)

%t t = {1, 3}; Do[AppendTo[t, t[[-1]] * t[[-2]] + 1], {n, 2, 14}] (* _T. D. Noe_, Jun 25 2012 *)

%o (Haskell)

%o a001056 n = a001056_list !! n

%o a001056_list = 1 : 3 : (map (+ 1 ) $

%o zipWith (*) a001056_list $ tail a001056_list)

%o -- _Reinhard Zumkeller_, Aug 15 2012

%o (PARI) m=13; v=concat([1,3], vector(m-2)); for(n=3, m, v[n]=v[n-1]*v[n-2] +1 ); v \\ _G. C. Greubel_, Sep 19 2019

%o (MAGMA) I:=[1,3]; [n le 2 select I[n] else Self(n-1)*Self(n-2) + 1: n in [1..13]]; // _G. C. Greubel_, Sep 19 2019

%o (Sage)

%o def a(n):

%o if (n==0): return 1

%o elif (n==1): return 3

%o else: return a(n-1)*a(n-2) + 1

%o [a(n) for n in (0..13)] # _G. C. Greubel_, Sep 19 2019

%o (GAP) a:=[1,3];; for n in [3..13] do a[n]:=a[n-1]*a[n-2]+1; od; a; # _G. C. Greubel_, Sep 19 2019

%Y Cf. A001622 (phi), A258112.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_, _R. K. Guy_

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Last modified June 4 08:18 EDT 2020. Contains 334825 sequences. (Running on oeis4.)