From Wikipedia:
Nelson Riddle was commissioned to write the instrumental theme when CBS decided to have a new song, rather than pay royalties for the Bobby Troup song “(Get Your Kicks on) Route 66.” Riddle’s theme, however, offers an unmistakable homage to the latter’s piano solo (as originally recorded by Nat King Cole) throughout the number.
I always liked The Stones version of the Bobby Troup tune. It got me driving a lot of Rt 66 in 1983 through Flagstaff Arizona, don’t forget Winona and so on
As I drove a car from California to the East Coast
Damn! I remember the music now...and the piano break. And it was a fresh piece of music...which included a deep bow to the original.
I imagine if you had asked Riddle (now deceased) who wrote the theme for Route 66, he might've replied "Well, actually, it was Bobby Troup...and I just fiddled with it in the arrangement."