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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.
(Formerly M0733)
3
1, 1, 1, 2, 3, 5, 11, 26, 81, 367, 2473, 32200, 939791, 80570391, 30341840591, 75749670168872, 2444729709746709953, 2298386861814452020993305, 185187471463742319884263934176321, 5618934645754484318302453706799174724040986 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
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
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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
EXAMPLE
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)
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 *)
PROG
(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
CROSSREFS
Sequence in context: A369495 A273755 A258804 * A009589 A098179 A055228
KEYWORD
nonn,easy
AUTHOR
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
STATUS
approved

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Last modified July 25 01:41 EDT 2024. Contains 374585 sequences. (Running on oeis4.)