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A074394 a(n) = a(n-1)*a(n-2) - a(n-3) with a(1) = 1, a(2) = 2, and a(3) = 3.

%I A074394 #34 Jan 21 2023 18:23:31
%S A074394 1,2,3,5,13,62,801,49649,39768787,1974480504962,78522694637486171445,
%T A074394 155041529758800625329015665441303,
%U A074394 12174278697379026530632791354719900462885271361687873
%N A074394 a(n) = a(n-1)*a(n-2) - a(n-3) with a(1) = 1, a(2) = 2, and a(3) = 3.
%C A074394 All consecutive quadruples are pairwise coprime. Multiples of 2 occur when n=2 mod 4, multiples of 3 when n=3 mod 4, multiples of 5 when n=4 mod 7, multiples of 7 when n=10 mod 14, multiples of 9 when n=7 or 11 mod 24, multiples of 10 when n=18 mod 28. Multiples of 4, 6 and 8 never occur.
%H A074394 Reinhard Zumkeller, <a href="/A074394/b074394.txt">Table of n, a(n) for n = 1..19</a>
%F A074394 Lim_{n->infinity} a(n+1)/a(n)^phi = 1, where phi is the golden ratio (1+sqrt(5))/2 = A001622. - _Benoit Cloitre_, Sep 26 2002
%F A074394 From _Jon E. Schoenfield_, May 13 2019: (Start)
%F A074394 It appears that, for n >= 2,
%F A074394   a(n) = ceiling(e^(c*phi^n - d/(-phi)^n))
%F A074394 where
%F A074394   phi = (1 + sqrt(5))/2
%F A074394     c = 0.230193077518834725477008740044380256486365499661...
%F A074394     d = 0.067704372842879037264190305626317036100889750046...
%F A074394 (End)
%e A074394 a(6) = a(5)*a(4) - a(3) = 13*5 - 3 = 62.
%t A074394 nxt[{a_,b_,c_}]:={b,c,b*c-a}; NestList[nxt,{1,2,3},15][[All,1]] (* _Harvey P. Dale_, Jan 21 2023 *)
%o A074394 (Haskell)
%o A074394 a074394 n = a074394_list !! (n-1)
%o A074394 a074394_list = 1 : 2 : 3 : zipWith (-)
%o A074394    (tail $ zipWith (*) (tail a074394_list) a074394_list) a074394_list
%o A074394 -- _Reinhard Zumkeller_, Mar 25 2015
%Y A074394 Cf. A001622, A022405, A061021, A072878, A072879, A072880.
%K A074394 nonn
%O A074394 1,2
%A A074394 _Henry Bottomley_, Sep 24 2002