To be frank, I am not able to find the formula for the below mentioned pattern(s) ( Yes its a pattern). Yet, there is a procedure / way / method / idea for everything. Here is one such thing.
Step 1 : For any talam, take the first avarthanam for your sollu ( Phrases)
step 2 : Sollu should be played 3 times and it should fit the avarthanam (with karvai or without karvai)
Note for step 2 : The sollu can be for 1 avarthanam or 2 avarthanams based on the length of talam
Step 3 : Add Tha thongu ( 1 3) at the end
Step 4 : Reduce the pharase by 1 mathrai and play three times and play tha thongu
Step 5 : Reduce the phrase again by 1 and play 3 times, and tha (it will land in samam for tham)
Example : For Roopaka thaalam
4 4 4 1 3 = 12 + 1 + 3= 16
3 3 3 1 3 = 9 + 1 + 3 = 13
2 2 2 1 = 6 + 1 = 7
16+13+7 = 36 ( will come for three avarthanam) can be played in Thisra nadai also. The same method can be applied for any talam, nadai, gathi.
Example for Kanda chapu :
6 . 6 . 6 1 3
5 . 5 . 5 1 3
4 . 4 . 4 1
Dot represents karvai. Its a simple pattern it can be improvised to any level.
Huge amount patterns can be found out to which i am not able to decipher a formula for this rather didnt really worked on how to decipher the same.
Moreover, the end part we play tha thom, which can be used for samam or for Atheetham 1 mathrirai and there are patterns for other eduppus also.
Critical Note : Formula and playing it aesthetically are completely different, Its is just for knowledge purpose, the final outcome of this pattern differs from person to person.