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%I #34 Jan 21 2023 18:23:31
%S 1,2,3,5,13,62,801,49649,39768787,1974480504962,78522694637486171445,
%T 155041529758800625329015665441303,
%U 12174278697379026530632791354719900462885271361687873
%N a(n) = a(n-1)*a(n-2) - a(n-3) with a(1) = 1, a(2) = 2, and a(3) = 3.
%C 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 Reinhard Zumkeller, <a href="/A074394/b074394.txt">Table of n, a(n) for n = 1..19</a>
%F 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 From _Jon E. Schoenfield_, May 13 2019: (Start)
%F It appears that, for n >= 2,
%F a(n) = ceiling(e^(c*phi^n - d/(-phi)^n))
%F where
%F phi = (1 + sqrt(5))/2
%F c = 0.230193077518834725477008740044380256486365499661...
%F d = 0.067704372842879037264190305626317036100889750046...
%F (End)
%e a(6) = a(5)*a(4) - a(3) = 13*5 - 3 = 62.
%t nxt[{a_,b_,c_}]:={b,c,b*c-a}; NestList[nxt,{1,2,3},15][[All,1]] (* _Harvey P. Dale_, Jan 21 2023 *)
%o (Haskell)
%o a074394 n = a074394_list !! (n-1)
%o a074394_list = 1 : 2 : 3 : zipWith (-)
%o (tail $ zipWith (*) (tail a074394_list) a074394_list) a074394_list
%o -- _Reinhard Zumkeller_, Mar 25 2015
%Y Cf. A001622, A022405, A061021, A072878, A072879, A072880.
%K nonn
%O 1,2
%A _Henry Bottomley_, Sep 24 2002
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