Let n>1 be an integer. The algebras of the title, which we abbreviate as algebras of type n, are infinite-dimensional graded Lie algebras L=⨁i=1∞Li, which are generated by an element of degree 1 and an element of degree n, and satisfy Li,L1=Li+1 for i≥n. Algebras of type 2 were classified by Caranti and Vaughan-Lee in 2000 over any field of odd characteristic. In this paper we lay the foundations for a classification of algebras of arbitrary type n, over fields of sufficiently large characteristic relative to n. Our main result describes precisely all possibilities for the first constituent length of an algebra of type n, which is a numerical invariant closely related to the dimension of its largest metabelian quotient.