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We continue our study of the Thresholding Greedy Algorithm when we restrictthe vectors involved in our approximations so that they either are supported onintervals of $\mathbb N$ or have constant coefficients. We introduce andcharacterize what we call consecutive greedy bases and provide newcharacterizations of almost greedy and squeeze symmetric Schauder bases.Moreover, we investigate some cases involving greedy-like properties withconstant 1 and study the related notion of Property (A,$\tau$).

On consecutive greedy and other greedy-like type of bases