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It is not clear whether the empty quandle is connected, so the sequence starts at order 1 instead of 0.

Quandles up to order 8 were determined first by Sam Nelson and co-authors (see references). Nelson's results were confirmed independently by the submitter, and extended to order 9.

A quandle (X,*) is commutative if a*b = b*a for all a,b in X. Every finite commutative quandle (X,*) is obtained from an odd order, commutative Moufang loop (X,+) where x*y = (1/2)(x+y). Thus a(n) is the number of isomorphism classes of commutative Moufang loops of order n if n is odd and is 0 if n is even. Commutative Moufang loops of order less than 81 are associative hence abelian groups. But, there are two non-associative commutative Moufang loops of order 81. Thus a(n) = number of isomorphism classes of abelian groups of odd order for n < 81 and a(81) = A000688(81) + 2 = 7. For proofs of these facts see, e.g., the papers below by Belousov, Nagy and Vojtchovský, and Glauberman.

A quandle is Latin if its multiplication table is a Latin square. A Latin quandle may be described as a left (or right) distributive quasigroup. Sherman Stein (see reference below) proved that a left distributive quasigroup of order n exists if and only if n is not of the form 4k + 2.

A rack or quandle X is medial (also called abelian) if the map X x X -> X defined by (x,y) -> y(x) is a rack homomorphism. Equivalently, the identity (xy)(uv)=(xu)(yv) holds for all elements x, y, u, and v in X.

a(n) is also the number of medial Legendrian racks of order n up to isomorphism; see Ta, "Equivalences of...," Theorem 1.1.

a(n) is also the number of medial generalized Legendrian quandles (also called GL-quandles or bi-Legendrian quandles) of order n up to isomorphism; see Ta, "Generalized Legendrian...," Theorem 5.5.

Generalized Legendrian racks, also called GL-racks or bi-Legendrian racks, are racks equipped with a rack automorphism that commutes with all inner rack automorphisms; see Ta, Definition 3.1 and Proposition 3.12. They are used to distinguish Legendrian links in contact three-space.

GL-racks are precisely virtual racks in which all inner rack automorphisms are virtual rack automorphisms; cf. Cattabriga and Nasybullov, Section 3.2.

A rack or quandle X is medial (also called abelian) if the map X x X -> X defined by (x,y) -> y(x) is a rack homomorphism. Equivalently, the identity (xy)(uv)=(xu)(yv) holds for all elements x, y, u, and v in X.

Generalized Legendrian racks, also called GL-racks or bi-Legendrian racks, are racks equipped with a rack automorphism that commutes with all inner rack automorphisms; see Ta, Definition 3.1 and Proposition 3.12. They are used to distinguish Legendrian links in contact three-space.

GL-racks are precisely virtual racks in which all inner rack automorphisms are virtual rack automorphisms; see Cattabriga and Nasybullov, Section 3.2.

A rack or quandle is medial if it satisfies the identity (xy)(uv) = (xu)(yv).

A Legendrian quandle is a pair (X,u) where X is a quandle and u is an involutory automorphism of X such that u(yx)=y(u(x)); see Ta, "Generalized Legendrian...," Corollary 3.13.

a(n) is also the number of medial racks X such that the kink map X -> X defined by x -> x(x) is an involution; see Ta, "Equivalences of...," Theorem 1.1.

A rack is involutory if it satisfies the identity y(yx) = x. In particular, involutory quandles are called kei.

a(n) is also the number of Legendrian kei (i.e., kei equipped with Legendrian structures) up to order n up to isomorphism; see Ta, Theorem 1.1.

a(n) is also the number of symmetric kei (i.e., kei equipped with good involutions) up to order n up to isomorphism; see Ta, "Equivalences of...," Corollary 1.3.

A virtual quandle is a quandle equipped with a distinguished quandle automorphism. Two virtual quandles (Q1,f1), (Q2,f2) are isomorphic if there exists a quandle isomorphism g: Q1 -> Q2 such that g*f1 = f2*g.