Search: id:A008404
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%I A008404
%S A008404 1,2,4,12,40,116,200,444,760,2160,4368,7852,12828,17252,19612,21104,
%T A008404 18276,15096,10240,6464,3536,2052,872,200,88,56,204
%N A008404 Number of Costas arrays of order n, counting rotations and flips as distinct.
%C A008404 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
%D A008404 CRC Handbook of Combinatorial Designs, C. Colbourn and J. Dinitz Eds., 
               1996, IV.7: Costas Arrays by Herbert Taylor (IV.7.6, page 259, Table 
               2.29).
%D A008404 CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 227.
%D A008404 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), 851-853.
%D A008404 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 
               0-7803-8234-X, pp. 260-265 (entries for orders 24 and 25).
%D A008404 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)
%H A008404 Ed Pegg, Jr., <a href="http://www.maa.org/editorial/mathgames/mathgames_11_15_04.html">
               Golomb Rulers</a>
%H A008404 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CostasArray.html">Costas Array</a>
%H A008404 K. Drakakis, <a href="http://www.costasarrays.org/Enumeration27TalkWeb.pdf">
               Results of the enumeration of Costas arrays oforder 27</a>.
%F A008404 There is no formula, recursion, or generating function for Costas arrays. 
               A number of number-theoretic 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
%e A008404 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
%e A008404 (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)
%o A008404 We use backtrack programming for exhaustive search and number-theoretic 
               generators for the Costas arrays that can be found that way. See 
               Beard et al., 2004 and IEEE Transacations AES, to appear.
%Y A008404 Cf. A001441.
%Y A008404 Sequence in context: A149846 A108532 A000940 this_sequence A099214 A126946 
               A113179
%Y A008404 Adjacent sequences: A008401 A008402 A008403 this_sequence A008405 A008406 
               A008407
%K A008404 nonn
%O A008404 1,2
%A A008404 N. J. A. Sloane (njas(AT)research.att.com).
%E A008404 More terms from James K Beard (jkbeard(AT)ieee.org), Nov 07 2005
%E A008404 a(27) (from the Drakakis link) sent by John Healy (johnjhealy(AT)gmail.com), 
               Jul 17 2008

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