Print Fibonacci series using recursion in bash with only 1 variable - bash

Question

I'd like to know how to print Fibonacci series using recursion in bash with only 1 variable.
From what I've done:
fib()
{
i=$1
if (( $i <= 1 ))
then echo 0
elif (( $i == 2 ))
then echo 1
else
echo $(( $(fib $(($i - 1)) ) + $(fib $(($i - 2)) ) ))
fi
}
echo $(fib $1)
I get the correct output of the final iteration, for example if I enter 10 I will get 34, but I'd like to print the whole sequence of numbers, i.e. all the iterations as well. How can I achieve that?
Another way I tried was by:
#!/bin/bash
arr[0]=0
arr[1]=1
for (( i=0; i<=10; i++ ))
do
echo -n "${arr[0]} "
arr[0]=$((${arr[0]} + ${arr[1]} ))
arr[1]=$((${arr[0]} - ${arr[1]} ))
done
echo ""
But obviously here I've used a for loop, but I don't want to use another variable.

Answer 1

Just for (my kind of) fun, this code prints the Fibonacci numbers from the 0th to the 92nd (as defined in Fibonacci number - Wikipedia) with a recursive function that uses no variables:
#! /bin/bash
function fib
{
echo ${3-0}
(($1 > 0)) && fib $(($1-1)) ${3-0} $((${2-1}+${3-0}))
}
fib 92
Some may claim that using the positional parameters ($1, $2, $3) for this is cheating, but then other solutions could be said to be using two variables ($i and $1).
The code takes under 0.01 seconds to run on my (oldish) Linux machine.
The code should work with numbers up to 92 with Bash version 3 or later on any platform. See Bash Number Limit?. Numbers higher than 93 will cause to code to produce garbage results due to arithmetic overflow.

Answer 2

Variables in bash are global by default. You need to make i local explicitly.
fib () {
local i
i=$1
if (( i <= 1 )); then
echo $i
else
echo $(( $(fib $((i-1)) ) + $(fib $((i - 2)) ) ))
fi
}
(Also, your base cases are a little off if you are starting with 0, and 2 need not be a base case; fib 2 can be derived from the base cases fib 0 and fib 1.)

Answer 3

If you want to print each fibonacci value from 1 to $n, I suggest:
fib_r() {
local i=$1
if (( i < 0 )); then
echo "Error: negative numbers not allowed" >&2
exit 1
elif (( i <= 1 )); then
echo $i
else
echo $(( $($FUNCNAME $((i - 1)) ) + $($FUNCNAME $((i - 2)) ) ))
fi
}
fib() {
local i
for (( i = 1; i <= $1; i++ )); do
fib_r $i
done
}
fib 10
outputs
0
1
1
2
3
5
8
13
21
34
It's still one variable, albeit one per function.
I use the bash variable $FUNCNAME in the recursive function so you don't have to hardcode the function name within itself. I got bit by not updating that line when I renamed the function.
Of course your performance will greatly improve if you cache the results: "fib 16" takes, on my VM, about 3.5 sec without caching and about 0.03 sec with caching.
fib_r() {
local i=$1
if (( i < 0 )); then
echo "Error: negative numbers not allowed" >&2
exit 1
elif [[ -n ${fib_cache[i]} ]]; then
echo "${fib_cache[i]}"
elif (( i <= 1 )); then
echo $i
else
echo $(( $( $FUNCNAME $((i - 1)) ) + $( $FUNCNAME $((i - 2)) ) ))
fi
}
fib_cache=()
fib() {
local i
for ((i=1; i<=$1; i++)); do
fib_cache[i]=$(fib_r $i)
echo "${fib_cache[i]}"
done
}