login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000952 Numbers k == 2 (mod 4) that are the orders of conference matrices.
(Formerly M1574 N0615)
8
2, 6, 10, 14, 18, 26, 30, 38, 42, 46, 50, 54, 62 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,hard,more,nice
AUTHOR
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 27 12:16 EST 2024. Contains 370375 sequences. (Running on oeis4.)