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

%I A001056 M2378 N0944 #43 Jan 09 2025 08:04:45
%S A001056 1,3,4,13,53,690,36571,25233991,922832284862,23286741570717144243,
%T A001056 21489756930695820973683319349467,
%U A001056 500426416062641238759467086706254193219790764168482,10754042042885415070816603338436200915110904821126871858491675028294447933424899095
%N A001056 a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.
%D A001056 Archimedeans Problems Drive, Eureka, 19 (1957), 13.
%D A001056 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001056 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001056 T. D. Noe, <a href="/A001056/b001056.txt">Table of n, a(n) for n = 0..17</a>
%H A001056 A. V. Aho and N. J. A. Sloane, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
%H A001056 A. V. Aho and N. J. A. Sloane, <a href="/A000058/a000058.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
%H A001056 <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>
%F A001056 a(n) ~ c^(phi^n), where c = A258112 = 1.7978784900091604813559508837..., phi = (1+sqrt(5))/2 = A001622. - _Vaclav Kotesovec_, Dec 17 2014
%p A001056 a:= proc (n) option remember;
%p A001056 if n=0 then 1
%p A001056 elif n=1 then 3
%p A001056 else a(n-1)*a(n-2) + 1
%p A001056 end if
%p A001056 end proc;
%p A001056 seq(a(n), n = 0..13); # _G. C. Greubel_, Sep 19 2019
%t A001056 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 A001056 t = {1, 3}; Do[AppendTo[t, t[[-1]] * t[[-2]] + 1], {n, 2, 14}] (* _T. D. Noe_, Jun 25 2012 *)
%o A001056 (Haskell)
%o A001056 a001056 n = a001056_list !! n
%o A001056 a001056_list = 1 : 3 : (map (+ 1 ) $
%o A001056                zipWith (*) a001056_list $ tail a001056_list)
%o A001056 -- _Reinhard Zumkeller_, Aug 15 2012
%o A001056 (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 A001056 (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 A001056 (Sage)
%o A001056 def a(n):
%o A001056     if (n==0): return 1
%o A001056     elif (n==1): return 3
%o A001056     else: return a(n-1)*a(n-2) + 1
%o A001056 [a(n) for n in (0..13)] # _G. C. Greubel_, Sep 19 2019
%o A001056 (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 A001056 Cf. A001622 (phi), A258112.
%K A001056 nonn,easy,nice
%O A001056 0,2
%A A001056 _N. J. A. Sloane_, _R. K. Guy_