Providence is nearly finished giving households new gray trash barrels to replace the large green ones that are now being used for single-stream recycling. The distribution of the barrels is expected to be completed citywide by the end of the year on Monday. The new 65-gallon gray barrels are for regular trash, while the 95-gallon ‘‘Big Green Cans’’ are now for recycling. Rhode Islanders no longer have to separate their recyclables in different receptacles. Providence has credited an increase in its recycling rate to the introduction of the green cans.