A006888 a(n) = a(n-1) + a(n-2)*a(n-3) for n > 2 with a(0) = a(1) = a(2) = 1.
1, 1, 1, 2, 3, 5, 11, 26, 81, 367, 2473, 32200, 939791, 80570391, 30341840591, 75749670168872, 2444729709746709953, 2298386861814452020993305, 185187471463742319884263934176321, 5618934645754484318302453706799174724040986
Offset: 0
Examples
From _Muniru A Asiru_, Jan 28 2018: (Start) a(3) = a(2) + a(1) * a(0) = 1 + 1 * 1 = 2. a(4) = a(3) + a(2) * a(1) = 2 + 1 * 1 = 3. a(5) = a(4) + a(3) * a(2) = 3 + 2 * 1 = 5. a(6) = a(5) + a(4) * a(3) = 5 + 3 * 2 = 11. a(7) = a(6) + a(5) * a(4) = 11 + 5 * 3 = 26. ... (End)
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..30
Programs
-
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
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Maple
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: seq(a(n), n=0..35); # Muniru A Asiru, Jan 28 2018
-
Mathematica
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 *) Nest[Append[#, Last[#] + Times @@ #[[-3 ;; -2]]] &, {1, 1, 1}, 17] (* Michael De Vlieger, Jan 23 2018 *) nxt[{a_,b_,c_}]:={b,c,c+b*a}; NestList[nxt,{1,1,1},20][[All,1]] (* Harvey P. Dale, Feb 03 2021 *)
Formula
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
Extensions
More terms from Michel ten Voorde Apr 11 2001
Typo in Mathematica code corrected by Vincenzo Librandi, Jun 09 2013
Definition clarified by Matthew Conroy, Jan 23 2018
Comments