

A008404


Number of Costas arrays of order n, counting rotations and flips as distinct.


18



1, 2, 4, 12, 40, 116, 200, 444, 760, 2160, 4368, 7852, 12828, 17252, 19612, 21104, 18276, 15096, 10240, 6464, 3536, 2052, 872, 200, 88, 56, 204, 712, 164
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OFFSET

1,2


COMMENTS

A Costas array is a permutation matrix that meets the Costas condition. The Costas condition has several equivalent definitions. One of them is that two square matrices defined from a Costas array, when overlaid with one of them offset by an integral number of rows and columns, will have no more than one 1 overlaid on another except when the number of shifts in both rows and columns is zero.  James K. Beard (jkbeard(AT)ieee.org), Nov 07 2005
Permutations such that each element in each row in the table of differences is unique, see second example. [Joerg Arndt, May 27 2012]


REFERENCES

James K. Beard, Jon C. Russo, Keith Erickson, Michael Moneleone and Mike Wright, Combinatoric collaboration on Costas arrays and radar applications, Proceedings of the IEEE 2004 Radar Conference, Apr 26, 2004, ISBN 078038234X, pp. 260265 (entries for orders 24 and 25).
James K. Beard, Jon C. Russo, Keith Erickson, Michael Moneleone and Mike Wright,"Costas Array Generation and Search Methodology," to appear in IEEE Transactions on Aerospace and Electronic Engineering. (Order 26)
CRC Handbook of Combinatorial Designs, C. Colbourn and J. Dinitz, Editors, 1996, IV.7: Costas Arrays by Herbert Taylor (IV.7.6, page 259, Table 2.29).
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 227.
K. Drakkis et al., On the disjointness of algebraically constructed Costas arrays, J. Algebra and Applications, 10 (2011), 219240.
J. Silverman, V. E. Vickers and J. M. Mooney, On the number of Costas arrays as a function of array size, Proc. IEEE, 76 (1988), 851853.


LINKS

Table of n, a(n) for n=1..29.
John P. Costas, A Study of Detection Waveforms Having Nearly Ideal RangeDoppler Ambiguity Properties, Proceedings of the IEEE, pp.9961009, August 1984.
K. Drakakis, Results of the enumeration of Costas arrays of order 27.
Ed Pegg, Jr., Golomb Rulers
Eric Weisstein's World of Mathematics, Costas Array


FORMULA

There is no formula, recursion, or generating function for Costas arrays. A number of numbertheoretic generators are known (see Golomb 1984, Beard 2004, etc.) but these do not generate all known Costas arrays of orders greater than twelve or so.  James K. Beard (jkbeard(AT)ieee.org), Nov 07 2005


EXAMPLE

A permutation matrix can be represented by a sequence of column indices, one for each row. A previously unknown Costas array of order 26 given this way is
(5, 8, 20, 16, 18, 15, 4, 25, 13, 19, 6, 10, 2, 0, 9, 24, 14, 21, 3, 23, 22, 7, 1, 11, 12, 17)
The permutation (2, 4, 8, 5, 10, 9, 7, 3, 6, 1) corresponds to a Costas array:
2 4 8 5 10 9 7 3 6 1 (Permutation: p(1), p(2), p(3), ..., p(n) )
2 4 3 5 1 2 4 3 5 (step1 differences: p(2)p(1), p(3)p(2), ... )
6 1 2 4 3 6 1 2 (step2 differences: p(3)p(1), p(4)p(2), ... )
3 6 1 2 7 3 6 (step3 differences: p(4)p(1), p(5)p(2), ... )
8 5 1 2 4 8 ( etc. )
7 3 5 1 9
5 1 2 4
1 2 7
4 3
1
This example is given in the Costas reference. [Joerg Arndt, May 27 2012]


PROG

We use backtrack programming for exhaustive search and numbertheoretic generators for the Costas arrays that can be found that way. See Beard et al., 2004 and IEEE Transactions AES, to appear.


CROSSREFS

Cf. A001441.
Sequence in context: A215071 A180487 A000940 * A170815 A170814 A170813
Adjacent sequences: A008401 A008402 A008403 * A008405 A008406 A008407


KEYWORD

nonn,more,hard


AUTHOR

N. J. A. Sloane.


EXTENSIONS

More terms from James K. Beard (jkbeard(AT)ieee.org), Nov 07 2005
a(27) (from the Drakakis link) sent by John Healy (johnjhealy(AT)gmail.com), Jul 17 2008
Added a(28) and a(29) (from http://www.costasarrays.org/), Joerg Arndt, May 27 2012.


STATUS

approved



