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The sequence A268289 from the On-Line Encyclopedia of Integer Sequences,namely the cumulated differences between the number of digits 1 and the numberof digits 0 in the binary expansion of consecutive integers, is studied here.This sequence happens to match a sequence of cardinalities of some specificsets. Furthermore, it can also be expressed by using the Takagi function. Thethree different definitions have their own properties and combining themtogether lead to some new identities.

Properties of the cumulated deficient binary digit sum