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

1, 1, 2, 3, 5, 8, 13, 20, 32, 51, 81, 129, 204, 324, 514, 816, 1295, 2056, 3263, 5180, 8222, 13052, 20718, 32888, 52206, 82872, 131551, 208824, 331488, 526204, 835297, 1325951, 2104816, 3341187, 5303804, 8419264, 13364749, 21215216, 33677057, 53458995

Offset

1,3

Comments

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.

Links

Table of n, a(n) for n=1..40.

Formula

a(n) ~ c * 2^(2*n/3), where c = 0.50182724761947676453167569419757096890286053854137516239835895319268638286015... - Vaclav Kotesovec, Dec 20 2018

Example

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.

Maple

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

Mathematica

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 *)

Prog

(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
(Java) public static int G(int n) {
if(n==1) {return 1; }
if(n==2) {return 1; }
return Math.round(2*Math.sqrt(G(n-1)*G(n-2))); //Recursive definition
} \\ James E Davis, Dec 03 2018

Crossrefs

Sequence in context: A266331 A055805 A023437 * A225394 A013985 A092834

Adjacent sequences: A322329 A322330 A322331 * A322333 A322334 A322335

Keyword

nonn

Author

James E Davis, Dec 03 2018

Status

approved