Google+ may have grabbed as many as 18 million users just three weeks after launch.
Calculating the number of Google+ users based on random surnames each day, Allen pointed to two days last week alone when more than 2 million users signed up in a single day.
Meanwhile, the Google+ iOS app, which launched just yesterday, has already climbed the charts to become the top free iPhone app in Apple's App Store.
Despite the huge numbers estimated by Allen, Google+ has actually seen a bit of a slowdown over the past few days following its initial growth spurt. In his research, Allen found that the social network captured only 763,000 new members on Monday, a growth rate of 4.47 percent and the lowest since Google started opening up invitations to the site July 6. Google+ is still invitation-only.
The recent slowdown may stem from the fact that Google hasn't marketed Google+ through any of its other channels or services. Once it does, Allen expects to see millions of people hop onto the service each day over a certain period of time. The company is reportedly trying to lure more celebrities to the site in an attempt to generate some buzz.
Allen, who is unrelated to Microsoft's co-founder of the same name, does concede that his model for calculating the number of users may have some "weaknesses." The search giant's CEO, Larry Page, announced Thursday that the site had captured more than 10 million members at the same time that Allen's research showed a total of 13 million.
"I suppose my model could be overstating the actual user count by 30-35 percent," Allen said in his Google+ post. "But if Google+ actually hit 10 million a day or two before the formal announcement, then my model may still be spot on."
The new iOS app, like its Android counterpart, lets people monitor posts from people they follow, incoming posts from people who follow them, and posts from people located in nearby geographic areas. Users can also send posts to their "circles," the groups of friends, family, and acquaintances that they can create.
Google spokeswoman Katie Watson confirmed to CNET that the initial version is available only for the iPhone but that "we are working quickly to roll out the Google+ mobile app to as many platforms and devices as possible." In the meantime, iPad and iPod Touch users can access the site directly via their mobile browser.
Updated 10:15 a.m. PT with comment from Google.