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A006888 a(n) = a(n-1) + a(n-2)*a(n-3) for n > 2 with a(0) = a(1) = a(2) = 1.

%I A006888 M0733 #53 Dec 29 2024 16:10:34
%S A006888 1,1,1,2,3,5,11,26,81,367,2473,32200,939791,80570391,30341840591,
%T A006888 75749670168872,2444729709746709953,2298386861814452020993305,
%U A006888 185187471463742319884263934176321,5618934645754484318302453706799174724040986
%N A006888 a(n) = a(n-1) + a(n-2)*a(n-3) for n > 2 with a(0) = a(1) = a(2) = 1.
%C A006888 Tends towards something like 1.60119...^(1.3247...^n) where 1.3247... = (1/2+sqrt(23/108))^(1/3)+(1/2-sqrt(23/108))^(1/3) is the smallest Pisot-Vijayaraghavan number A060006. Any four consecutive terms are pairwise coprime. - _Henry Bottomley_, Sep 25 2002
%D A006888 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006888 Vincenzo Librandi, <a href="/A006888/b006888.txt">Table of n, a(n) for n = 0..30</a>
%F A006888 Limit_{n->infinity} a(n)/(a(n-1)*a(n-5)) = 1 agrees with lim_{n->infinity} a(n) = c^(P^n) (c=1.60119..., P=PisotV) since PisotV is real root of x^3-x-1 and thus a root of x^5-x^4-1 because x^5-x^4-1 = (x^3-x-1)*(x^2-x+1) and c^(P^n)/(c^(P^(n-1))*c^(P^(n-5))) = c^(P^(n-5)*(P^5-P^4-1)). - _Gerald McGarvey_, Aug 14 2004
%e A006888 From _Muniru A Asiru_, Jan 28 2018: (Start)
%e A006888 a(3) = a(2) + a(1) * a(0) = 1 + 1 * 1 = 2.
%e A006888 a(4) = a(3) + a(2) * a(1) = 2 + 1 * 1 = 3.
%e A006888 a(5) = a(4) + a(3) * a(2) = 3 + 2 * 1 = 5.
%e A006888 a(6) = a(5) + a(4) * a(3) = 5 + 3 * 2 = 11.
%e A006888 a(7) = a(6) + a(5) * a(4) = 11 + 5 * 3 = 26.
%e A006888 ...
%e A006888 (End)
%p A006888 a := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n=2 then 1 elif n>=3 then procname(n-1) + procname(n-2) * procname(n-3) fi; end:
%p A006888 seq(a(n), n=0..35); # _Muniru A Asiru_, Jan 28 2018
%t A006888 a=1;b=1;c=1;lst={a,b,c};Do[d=a*b+c;AppendTo[lst,d];a=b;b=c;c=d,{n,2*4!}];lst  (* _Vladimir Joseph Stephan Orlovsky_, Sep 13 2009 *)
%t A006888 Nest[Append[#, Last[#] + Times @@ #[[-3 ;; -2]]] &, {1, 1, 1}, 17] (* _Michael De Vlieger_, Jan 23 2018 *)
%t A006888 nxt[{a_,b_,c_}]:={b,c,c+b*a}; NestList[nxt,{1,1,1},20][[All,1]] (* _Harvey P. Dale_, Feb 03 2021 *)
%o A006888 (GAP) a := [1,1,1];; for n in [4..35] do a[n] := a[n-1] + a[n-2] * a[n-3]; od; a; # _Muniru A Asiru_, Jan 28 2018
%K A006888 nonn,easy
%O A006888 0,4
%A A006888 _Robert Munafo_
%E A006888 More terms from _Michel ten Voorde_ Apr 11 2001
%E A006888 Typo in Mathematica code corrected by _Vincenzo Librandi_, Jun 09 2013
%E A006888 Definition clarified by _Matthew Conroy_, Jan 23 2018