Reading the Oklahoma City Thunder's page on Statistical Analysis Primer, by Kevin Pelton of Supersonics.com,
The TSP metric is discussed as follows:
Shooting Efficiency - If there is an on-base percentage in the NBA - a statistic that has traditionally been undervalued - it would probably be some measure of a player's efficiency in scoring points. There's a stereotype that all statistical analysts think Allen Iverson is a bad player due to his low shooting percentage that is untrue because Iverson's ability to create shots and get his teammates better looks is valuable. Still, being efficient with your shots is very important. The two most common ways of measuring the concept of shooting efficiency are Effective Field-Goal Percentage (eFG%) and what this site calls True Shooting Percentage (TS%).
Effective Field-Goal Percentage was popularized by current L.A. Clippers Coach Mike Dunleavy and the Rick Barry's Pro Basketball Bible series. It adjusts for the added value of three-pointers by counting them as 1.5 field goals, thus make it more fair to three-point shooters than field-goal percentage.
eFG% = (FGM + .5*3PM)/FGA
True Shooting Percentage goes a step further by factoring in a player's performance at the free-throw line and considering their efficiency on all types of shots.
TS% = Pts/(2*(FGA + (.44*FTA)))
Former Sonics guard Brent Barry - Rick's son - led the NBA in both categories in 2006-07, posting a 62.6% effective field-goal percentage and a 66.6% True Shooting Percentage. Barry has led the NBA in True Shooting Percentage three times, including twice while in Seattle. Rashard Lewis (58.7%) was the most efficient Sonics shooter by True Shooting Percentage.
This matches what you have as well as many other sites, including Wikipedia's page on True Shooting Percentage.
As far as the 44% goes, I finally found a page on fansided.com, named Nylon Calculus 101: True Shooting Percentage by Justin.
In it he mentions:
What’s with the .44?
You want to convert free throws into a shot attempt equivalent. Since players usually take two free throws at a time, the coefficient should be around 0.50. But we have to account for and-1’s, technical free throws, flagrant foul free throws, and 3PT free throws (three in a row.) Thus, based on testing, the 0.44 coefficient is used for the NBA, and it’s surprisingly accurate.
He goes on to discuss a perhaps slightly more accurate way to come up with an exact percentage multiplier, but says that using 44% (0.44) for FTAs is accurate enough for general purposes.