%I #14 Feb 18 2022 21:36:27

%S 1,1,5,33,675,65066,41160471

%N a(n) is the number of nonisomorphic flexible left-right-alternative magmas with n elements. That is, a(n) is the number of nonisomorphic magmas with n elements which satisfy all of the identities x(yx) = (xy)x, (xx)y = x(xy), and x(yy) = (xy)y (for all x and y).

%C 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).

%e There are 10 nonisomorphic magmas with 2 elements, 5 of which are flexible and left-right-alternative, so a(2) = 5.

%e Similarly there are 3330 nonisomorphic magmas with 3 elements, 33 of which satisfy all of (xy)x = x(yx), (xx)y = x(xy), and x(yy) = (xy)y for all x and y, so a(3) = 33.

%Y Cf. A001329 (magmas), A350873 (flexible magmas), A350874 (left/right-alternative magmas), A350875 (left-right-alternative magmas).

%K nonn,hard,more

%O 0,3

%A _Joel Brennan_, Jan 23 2022

%E a(5)-a(6) from _Andrew Howroyd_, Jan 25 2022