Derive the respective closed$-$form T $($n$)$ for the number of times the function Bar is called.

$($a$)$

$1$: function GloriousCount$($n : Z$+)$

$2$: count $0$

$3$: for i from $1$ to n $1$ do

$4$: for j from $0$ to n $1$ do

$5$: count $$ Bar$($i $+$ j$)$

$6$: end for

$7$: end for

$8$: for k from $0$ to n$21$ do

$9$: count $$ Bar$($k$)$

$10$: end for

$11$: return count

$12$: end function

$($b$)$

$1$: function SuperbCount$($n : Z$+$ power of $2)$

$2$: count $0$

$3$: for i from n to $1$ by dividing by $2$ do

$4$: for j from $1$ to i do

$5$: count $$ Bar$($i $+$ j$)$

$6$: end for

$7$: end for

$8$: return count

$9$: end function

$($c$)$

$1$: function FantasticCount$($n : Z$+)$

$2$: count $0$

$3$: for i from $0$ to n$31$ do

$4$: for j from i to n do

$5$: count $$ Bar$($i $+$ j$)$

$6$: end for

$7$: end for

$8$: return count

$9$: end function

$3$