"after thinking for a while" is correct and usable in written English. You can use it when you want to explain that, after a period of consideration, you came to a certain conclusion. For example: "After thinking for a while, I decided to go back to school and pursue a degree in engineering."
After thinking for a while Mr Clegg said it had been "quite an expensive one", recalling that he had gone to a "a burger place called Byron" after taking his children to the cinema.
After thinking for a while, Johnny exclaimed, "I've got it!
After thinking for a bit, Busque admits that there is.
He'd been thinking for a while about retiring.
So he sat there thinking for a while.
"You know," he said, "I was thinking for a while of leaving the city".
It keeps them thinking for a while.
Keep thinking for a while, you might eventually remember.
Give your brain a break, and stop thinking for a while.
In the end, after all, what they are selling is meant to let you kick back and stop thinking for a little while.
But after thinking about it for a while, Chan has figured out a way to simplify the math.
I love the desktop app, it’s always running on my Mac. Ludwig is the best English buddy, it answers my 100 queries per day and stays cool.
Cristina Valenza
Retail Lead Linguist @ Apple Inc.