The usual convergence of sequences has many generalizations with the aim of providing deeper insights into the summability theory. In this paper, following a very recent and new approach, we introduce the notion of I3 and I*3−convergence of triple sequences in neutrosophic normed spaces mainly as a generalization of statistical convergence of triple sequences. We investigate a few fundamental properties and study the relationship between the two notions. We also introduce and investigate the concept of I3 and I*3−Cauchy sequence of triple sequences and show that the condition (AP3) plays a crucial role to study the interrelationship between them.Publisher's Versio