login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000952 Numbers n == 2 (mod 4) that are the orders of conference matrices.
(Formerly M1574 N0615)
5
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 n is an n X n {-1,0,+1} matrix A such that A A' = (n-1)I.

If n == 2 (mod 4) then a necessary condition is that n-1 is a sum of 2 squares. It is conjectured that this condition is also sufficient. If n == 2 mod 4 and n-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.

N. A. Balonin  and Jennifer Seberry, A review and new symmetric conference matrices, http://ro.uow.edu.au/cgi/viewcontent.cgi?article=3757&context=eispapers, 2014.

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

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

Sequence in context: A250198 A260084 A194282 * A281702 A281703 A180216

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 if a 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 | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 28 11:45 EDT 2017. Contains 284186 sequences.