A095025 Number of inequivalent cyclic difference sets with n elements.

1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 0, 2, 1, 0, 1, 2, 0, 1, 1, 1, 1, 0, 2, 1, 1, 3, 1, 3, 0, 1, 0, 0, 1, 1, 4, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 6, 0, 2, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 3

Author

Hugo Pfoertner, May 27 2004

Keywords

• nonn

Comments

A (v,k,lambda) cyclic difference set is a subset D = {d_1, d_2, ..., d_k} of the integers modulo v such that {1, 2, ..., v-1} can each be represented as a difference (d_i-d_j) modulo v in exactly lambda different ways.

If D is a cyclic difference set, then D+a and u*D are again cyclic difference sets, for any a and any invertible u, cf. examples. Therefore this sequence counts only the equivalence classes of sets modulo such transformations. - M. F. Hasler, Jul 30 2021

Examples

			a(3) = 1 corresponds to the (7,3,1) set D = {1, 2, 4}: Each of {1, ..., 6} (mod 7) has exactly 1 representation as difference of two elements in D: 1 = 2 - 1; 2 = 4 - 2; 3 = 4 - 1; 4 == 1 - 4 (mod 7); 5 == 2 - 4 (mod 7); 6 == 1 - 2 (mod 7). The "shifted" sets {2, 3, 5}, {3, 4, 6}, {0, 4, 5}, {1, 5, 6}, {0, 2, 6}, {0, 1, 3} and -D == {3, 5, 6} == 3*D = -2*D and shifted variants of this set automatically also yield all elements of {1, ..., 6} (mod 7) exactly once as difference of two elements, but these "equivalent" variants are not counted separately.
a(4) = 1 corresponds to the (13,4,1) set D' = {0, 1, 3, 9}: again, each of {1, ..., 12} have exactly one representation as x - y (mod 13) with x, y in D'.
a(5) = 2 because there are two cyclic difference sets of length 5: The (v,k,lambda)=(11,5,2) set A095028 = {1, 3, 4, 5, 9} and the (21,5,1) set A095029 = {3, 6, 7, 12, 14}.
		

Links

• Ralf Goertz, Coprime ordering of cyclic planar difference sets, Discrete Mathematics, Volume 309, Issue 16, 2009, Pages 5248-5252.
• Dan Gordon, Search Form for Difference Sets
• Dan Gordon, La Jolla Difference Set Repository [from Internet Archive Wayback Machine]
• Dan Gordon, List of Cyclic Difference Sets [from Internet Archive Wayback Machine]
• Dan Gordon and Len Baumert, Papers on Difference Sets

Crossrefs

Cf. A095029 - A095047: examples of cyclic difference set with 5 <= k <= 20.

Extensions

Second example corrected by an anonymous reader - N. J. A. Sloane, Jul 19 2021

Definition clarified by M. F. Hasler, Jul 30 2021