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A006888
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a(n) = a(n-1) + a(n-2)*a(n-3) for n > 2 with a(0) = a(1) = a(2) = 1.
(Formerly M0733)
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3
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1, 1, 1, 2, 3, 5, 11, 26, 81, 367, 2473, 32200, 939791, 80570391, 30341840591, 75749670168872, 2444729709746709953, 2298386861814452020993305, 185187471463742319884263934176321, 5618934645754484318302453706799174724040986
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OFFSET
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0,4
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
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EXAMPLE
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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)
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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