cp's OEIS Frontend

Equivalently (by symmetry), a(n) also equals the number of nonisomorphic right-alternative magmas with n elements (that is, magmas satisfying the identity x(yy) = (xy)y for all x and y).

Compare with A350876, whose terms are smaller (for n > 2) - this means that the left and right alternative identities (xx)y = x(xy) and x(yy) = (xy)y do not imply the flexible identity (xy)x = x(yx) for magmas. This is in contrast to the situation for non-associative rings, where left-right-alternativity implies flexibility (due to the additional additive structure).

a(n) = A350874(n) for n <= 2, i.e., a magma with (zero, one or) two elements which is left (resp., right) alternative is also right (resp., left) alternative.

Compare A350873 and A350875, which are the numbers of flexible magmas with n elements and left-right-alternative magmas with n elements (up to isomorphism). The fact that a(n) < A350875(n) for n >= 3 means that left-right-alternativity for magmas (the identities (xx)y = x(xy) and x(yy) = (xy)y) does not imply flexibility (x(yx) = (xy)x). This is in contrast to the situation for non-associative rings, where this implication does hold (due to the additional additive structure).