What's the solution to this puzzle? I saw this on Instagram with no solution and was wondering what the answer is.  I got $33$.
$$1+4=5$$
$$2+5=12$$
$$3+6=21$$
$$8+11=?$$

$a_n + b_n = c_n$
should in fact be interpreted as
$a_n \cdot b_n + n = c_n$
with
$b_n = a_n + 3$ 
and the following recursive relation for $a_n$
$a_n = a_{n-1} \cdot a_{n-2} + (n-2),\:\:$ $n \geq 3$
$a_0 = 0, \:\:\:\:a_1 = 1, \:\:\:\:a_2 =2$ 

$a_1 \cdot b_1 + 1 = c_1$
$a_{2} \cdot b_{2} + 2 = c_{2}$
$a_{3} \cdot b_{3} + 3 = c_{3}$
$a_{4} \cdot b_{4} + 4 = c_{4}$
give us 
$1+4=5$
$2+5=12$
$3+6=21$
$8+11=c_4$
and
$c_{4}$  
$= a_{4} \cdot b_{4} + 4 = 92$
or to demonstrate the recursion
$c_{4}$ 
$= (a_{3} \cdot a_{2} + 2) \cdot b_{4} + 4 $
$= (3 \cdot 2 + 2) \cdot 11 + 4$ 
$= 8 \cdot 11 + 4 = 92$

This results in the following sequence
$1+4=5$
$2+5=12$
$3+6=21$
$8+11=92$
$200 + 203 = 40605$
$4824 + 4827 = 23285454$
$964808 + 964811 = 930857371296$
$\vdots$ $\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\vdots$ 

Im pretty sure this is what they had in mind :)
