Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 adj. Following in order or time; subsequent.
 adj. Following as a result; consequent.
 n. A result; a consequence.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. That comes after in time or order; subsequent.
 adj. That follows on as a result, conclusion etc.; consequent to, on, upon.
 adj. Recurring in succession or as a series; successive, consecutive.
 n. Something that follows in a given sequence.
 n. An element of a sequence, usually a sequence in which every entry is an axiom or can be inferred from previous elements.
 n. A follower.
from the GNU version of the Collaborative International Dictionary of English
 adj. Following; succeeding; in continuance.
 adj. Following as an effect; consequent.
 n. A follower.
 n. That which follows as a result; a sequence.
from The Century Dictionary and Cyclopedia
 Continuing in the same course or order; following; succeeding.
 Following by natural or logical consequence.
 n. A follower.
 n. A sequence or sequel; that which follows as a result.
 n. That which follows by an observed order of succession: used, in opposition to antecedent, where one wishes to avoid the implication of the relation of effect to cause that would be conveyed by the use of consequent.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 adj. in regular succession without gaps
 adj. following or accompanying as a consequence
Etymologies
Examples

He therefore invented another logical calculus that he called sequent calculus (Sequenzenkalkul, literally “calculus of sequences”) and made it the central topic of his thesis.

The ten thralls answered him with one mouth and in sequent words, saying, Whatso thou biddest us,

If at any stage of this “reduction process” the conclusion of a sequent is a compound formula, you have to consider any conjunct or any instance of universal quantification as a possible conclusion.

Every body know their truth by sequent 12 on Thursday, Sep 3, 2009 at 2: 00: 12 AM

Inside the magazine, she is shown wearing a range of outfits, including a sequent top and leather pants.

Ketonen's idea was to define a system of proof search: one starts from a given sequent to be derived, chooses a formula in it, and writes the premisses of a rule that can conclude the given sequent.

A cut with such a sequent as one premiss has the other premiss equal to the conclusion and can therefore be deleted.

Failure of the aims of Hilbert through GÃ¶del's incompleteness theorems; Gentzen's creation of the two main types of logical systems of contemporary proof theory, natural deduction and sequent calculus

Ketonen wanted to formulate Skolem's formal rules of proof within sequent calculus.

With Kleene's book, Gentzen's sequent calculi became generally known and accessible.
yarb commented on the word sequent
He learned the wheelwright's trade, and really that seemed all there was to tell. The rest had been calm and sequent progressionsteady employment as a journeyman first; then marriage and a house and lot; the modest start as a master; the move to Octavius and cheap lumber; the growth of his business, always marked of late years stupendousall following naturally, easily, one thing out of another.
 Harold Frederic, The Damnation of Theron Ware, ch. 9
August 1, 2008