A000952 Numbers k == 2 (mod 4) that are the orders of conference matrices. (Formerly M1574 N0615)

2, 6, 10, 14, 18, 26, 30, 38, 42, 46, 50, 54, 62

Offset

1,1

Comments

A conference matrix of order k is a k X k {-1,0,+1} matrix A such that A A' = (k-1)I.

If k == 2 (mod 4) then a necessary condition is that k-1 is a sum of 2 squares (A286636). It is conjectured that this condition is also sufficient. If k == 2 (mod 4) and k-1 is a prime or prime power the condition is automatically satisfied.

References

V. Belevitch, Conference matrices and Hadamard matrices, Ann. Soc. Scientifique Bruxelles, 82 (I) (1968), 13-32.

CRC Handbook of Combinatorial Designs, 1996, Chapter 52.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 56.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..13.

Joerg Arndt, Some relevant PARI/GP programs

N. A. Balonin and Jennifer Seberry, A review and new symmetric conference matrices, 2014.

Nikolay Balonin, Mikhail Sergeev and Anton Vostrikov, Prime Fermat numbers and maximum determinant matrix conjecture, Information and Control Systems (2020) No. 2, 2-9. (Abstract in Russian, English translation available on page)

Wikipedia, Conference matrix.

Example

The essentially unique conference matrix of order 6:
   0 +1 +1 +1 +1 +1
  +1  0 +1 -1 -1 +1
  +1 +1  0 +1 -1 -1
  +1 -1 +1  0 +1 -1
  +1 -1 -1 +1  0 +1
  +1 +1 -1 -1 +1  0

Crossrefs

Subsequence of A016825.

Cf. A286636.

Sequence in context: A250198 A260084 A194282 * A286636 A291783 A322992

Adjacent sequences: A000949 A000950 A000951 * A000953 A000954 A000955

Keyword

nonn,hard,more,nice

Author

N. J. A. Sloane

Extensions

66 seems to be the smallest order for which it is not known whether a conference matrix exists. Since 65 is the sum of two squares, according to the conjecture, 66 should be the next term.

Edited by N. J. A. Sloane, Mar 13 2008, Mar 16 2008, May 22 2014

Status

approved