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%I #23 Sep 08 2022 08:46:13
%S 1,2,3,3,5,7,11,20,40,94,275,1017,5128,37871,444415,9290130,395420005,
%T 40404949540,12183091294648,13951642918891149,82872169787001239679,
%U 3753148776564192982863648,2083123034674803589767277778237,25454214863632278822694894280883452911
%N a(1)=1, a(2)=2, a(3)=3, a(n) = ceiling(sqrt(a(n-1)*a(n-2)*a(n-3))), n>3.
%H Harvey P. Dale, <a href="/A258967/b258967.txt">Table of n, a(n) for n = 1..39</a>
%F a(n) is approximately k^(c^n), where c is the real root of x^3 - (x^2 + x + 1)/2 = 0 equal to (1 + (64 - 3*sqrt(417))^(1/3) + (64 + 3*sqrt(417))^(1/3))/6, and k is approximately 1.7450496...
%e a(4) = ceiling(sqrt(1*2*3)) = 3;
%e a(5) = ceiling(sqrt(2*3*3)) = 5;
%e a(6) = ceiling(sqrt(3*3*5)) = 7.
%t RecurrenceTable[{a[n] == Ceiling[Sqrt[a[n - 1] a[n - 2] a[n - 3]]], a[1] == 1, a[2] == 2, a[3] == 3}, a, {n, 1, 23}] (* _Michael De Vlieger_, Jul 02 2015 *)
%t a[1] = 1; a[2] = 2; a[3] = 3; a[n_] := a[n] = Ceiling[ Sqrt[ a[n - 1]*a[n - 2]*a[n - 3]]]; Array[a, 23] (* _Robert G. Wilson v_, Aug 12 2015 *)
%t nxt[{a_,b_,c_}]:={b,c,Ceiling[Sqrt[a*b*c]]}; NestList[nxt,{1,2,3},30][[All,1]] (* _Harvey P. Dale_, Sep 09 2021 *)
%o (Magma) I:=[1, 2, 3]; [n le 3 select I[n] else Ceiling(Sqrt(Self(n-1)*Self(n-2)*Self(n-3))): n in [1..23]];
%o (PARI) first(m)={my(v=vector(m));v[1]=1;v[2]=2;v[3]=3;for(i=4,m,v[i]=ceil(sqrt(v[i-1]*v[i-2]*v[i-3])));v;} \\ _Anders Hellström_, Aug 20 2015
%Y Cf. A258875, A254400.
%K nonn
%O 1,2
%A _Morris Neene_, Jun 15 2015
%E Corrected and extended by _Harvey P. Dale_, Sep 09 2021
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