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A322332 'Geobonnaci' sequence: a(1)=a(2)=1, thereafter a(n) = round( 2 * sqrt(a(n-1) * a(n-2)) ).

%I A322332 #33 Dec 04 2024 16:46:02
%S A322332 1,1,2,3,5,8,13,20,32,51,81,129,204,324,514,816,1295,2056,3263,5180,
%T A322332 8222,13052,20718,32888,52206,82872,131551,208824,331488,526204,
%U A322332 835297,1325951,2104816,3341187,5303804,8419264,13364749,21215216,33677057,53458995
%N A322332 'Geobonnaci' sequence: a(1)=a(2)=1, thereafter a(n) = round( 2 * sqrt(a(n-1) * a(n-2)) ).
%C A322332 Named because each term is the nearest integer to twice the geometric mean of the previous two terms. This is similar to the Fibonacci sequence, where each term is twice the arithmetic mean of the previous two terms. In fact, the early terms mirror the Fibonacci sequence.
%H A322332 Harvey P. Dale, <a href="/A322332/b322332.txt">Table of n, a(n) for n = 1..1000</a>
%F A322332 a(n) ~ c * 2^(2*n/3), where c = 0.50182724761947676453167569419757096890286053854137516239835895319268638286015... - _Vaclav Kotesovec_, Dec 20 2018
%e A322332 For n=6, a(5) is 5 and a(4) is 3. 3 * 5 is 15 and twice the square root of 15 is just above 7.745. This rounds to 8, so a(6) is 8.
%p A322332 a:=proc(n) option remember: `if`(n<3,1,round(2*sqrt(a(n-1)*a(n-2)))) end: seq(a(n),n=1..50); # _Muniru A Asiru_, Dec 20 2018
%t A322332 a[1] =1 ; a[2] = 1; a[n_] := a[n] = Round[2 * Sqrt[a[n-1] * a[n-2]]]; Array[a, 40] (* _Amiram Eldar_, Dec 04 2018 *)
%t A322332 nxt[{a_,b_}]:={b,Round[2*Sqrt[a*b]]}; NestList[nxt,{1,1},40][[;;,1]] (* _Harvey P. Dale_, Dec 04 2024 *)
%o A322332 (PARI) seq(n)={my(v=vector(n)); v[1]=v[2]=1; for(n=3, n, v[n]=round(2*sqrt(v[n-1]*v[n-2]))); v} \\ _Andrew Howroyd_, Dec 03 2018
%o A322332 (Java) public static int G(int n) {
%o A322332 if(n==1) {return 1;}
%o A322332 if(n==2) {return 1;}
%o A322332 return Math.round(2*Math.sqrt(G(n-1)*G(n-2))); //Recursive definition
%o A322332 } \\ _James E Davis_, Dec 03 2018
%K A322332 nonn
%O A322332 1,3
%A A322332 _James E Davis_, Dec 03 2018