It took ??1.6m in its opening weekend, but could not topple Doctor Strange or Trolls from the first two spots.
The Marvel film starring Benedict Cumberbatch topped the chart for a second week, taking ??3.4m, while Trolls stayed at number two, with ??2m.
There was another new entry for A Street Cat Named Bob, based on the true story of a homeless man and his cat.
It entered the UK box office at number four taking ??990,000.
The Accountant sees Ben Affleck appear as Christian Wolff, who uses his accountancy office as a front for his work for criminal organisations.
The thriller, also starring Anna Kendrick, topped the US box office on its opening weekend there.
Nocturnal Animals, by fashion designer and director Tom Ford, was another new entry in the top five, taking ??760,000.
The film, starring Amy Adams and Jake Gyllenhaal, is the second from Ford, after A Single Man, which was released in 2009.
Elsewhere in the top 10, The Light Between Oceans - a period romance starring Michael Fassbender and Alicia Vikander - entered the chart at number six, taking ??730,000 on its opening weekend.
The Girl on the Train stayed in the top 10 for a fifth week, at number seven, followed by Jack Reacher: Never Go Back, Storks and Bridget Jones's Baby, which is now in its eighth week.
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The Accountant has entered the UK box office at number three, making it the highest new entry this week.