What sequence goes in this pattern 1, 2, 5, 13, 34
Do you know of the Fibonnaci Sequence?
$\displaystyle F_{n + 2} = F_{n + 1} + F_n$, with $\displaystyle F_0 = 0$ and $\displaystyle F_1 = 1$.
In other words, each term is found by adding the previous two.
$\displaystyle 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$.
Can you see that your sequence is the Fibonnaci sequence with each even subscript removed (assuming you start at $\displaystyle F_0$?
Hello, Liu997!
I too recognized the alternate terms of the Fibonacci sequence,What sequence goes in this pattern? .$\displaystyle 1,\: 2,\: 5,\: 13,\: 34,\:\hdots$
. . but what if I didn't?
The expected answer is probably: .$\displaystyle a_{n+1}\;=\;3a_n - a_{n-1}$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
The "other" terms of the Fibonacci sequence has the same recurrence:
. . $\displaystyle 1,\:3,\:8,\:21,\:55,\:144,\:\hdots \qquad a_{n+1} \:=\:3a_n - a_{n-1}$
Why am I surprised?
.