late 14c., in church music, a composition said or sung after the Alleluia and before the Gospel, from Old French sequence "answering verses" (13c.) and directly from Medieval Latin sequentia "a following, a succession," from Latin sequentem (nominative sequens), present participle of sequi "to follow" (from PIE root *sekw- (1) "to follow").
In Church use, a partial loan-translation of Greek akolouthia, from akolouthos "following." By 1570s in the general sense of "a series of things following in a certain order, a succession," also in cards, "a run of three or more consecutive numbers of the same suit." By 1580s as "order of succession." In biochemistry in reference to nucleic acids, by 1959.
"arrange in a sequence," 1954, in computing, from sequence (n.). In biochemistry by 1970. Related: Sequenced; sequencing.