%I A085000
%S A085000 1,10,412,40800,6839492,1865999570,762150368499
%N A085000 Maximal determinant of an n X n matrix using the integers 1 to n^2.
%C A085000 Known lower bounds for the next terms are a(8)>=440943507851753, a(9)>
               =3.46254605664*10^17 and a(10)>=3.56944784623*10^20, found by Hermann 
               Jurksch (jurksch(AT)elektron-bbs.de). The corresponding matrices 
               are provided in the program given at the Pfoertner link. - Hugo Pfoertner 
               (hugo(AT)pfoertner.org), Jan 22 2008
%H A085000 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a085000.txt">
               Maximal determinant of matrix with elements 1..n.</a> FORTRAN program.
%H A085000 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SquareMatrix.html">Square Matrix</a>
%e A085000 The following 3 X 3 matrix is one of 36 whose determinant is 412 (there 
               are also 36 whose determinant is -412):
%e A085000 9 3 5
%e A085000 4 8 1
%e A085000 2 6 7
%e A085000 Results from a specially-adapted hill-climbing algorithm strongly suggest 
               that a(5) = 6839492. a(6) is at least 1862125166. Heuristically, 
               a(n) is approximately 0.44 * n^(2.06*n), suggesting that a(7) is 
               close to 6.8 * 10^11. - Tim Paulden (timmy(AT)cantab.net), Sep 21 
               2003
%e A085000 a(5) confirmed by Robert Israel (israel(at) math.ubc.ca) and Hugo Pfoertner. 
               A corresponding matrix is ((25 15 9 11 4) (7 24 14 3 17) (6 12 23 
               20 5) (10 13 2 22 19) (16 1 18 8 21) ) - Hugo Pfoertner (hugo(AT) 
               pfoertner.org), Sep 23 2003
%e A085000 a(6) found with FORTRAN program given at Pfoertner link. A corresponding 
               matrix is ((36 24 21 17 5 8) ( 3 35 25 15 23 11) (13 7 34 16 10 31) 
               (14 22 2 33 12 28) (20 4 19 29 32 6) (26 18 9 1 30 27) ) - Hugo Pfoertner 
               (hugo(AT) pfoertner.org), Sep 23 2003
%e A085000 a(7) is the determinant of the matrix ((46 42 15 2 27 24 18) (9 48 36 
               30 7 14 31) (39 11 44 34 13 29 5) (26 22 17 41 47 1 21) (20 8 40 
               6 33 23 45) (4 28 19 25 38 49 12) (32 16 3 37 10 35 43)). Although 
               no proof for the optimality of a(7) is available, the results of 
               an extensive computational search make the existence of a better 
               solution extremely unlikely. A total of approximately 15 CPU years 
               on SGI Origin 3000 and of 3.8 CPU years on SGI Altix 3000 computers 
               was used for this result.
%t A085000 Needs["DiscreteMath`Combinatorica`"]; n=3; n2=n^2; dMax=0; mMax={}; p=Range[n2]; 
               Do[m=Partition[p, n]; d=Det[m]; If[d>dMax, dMax=d; mMax=m]; p=NextPermutation[p], 
               {k, n2!}]; {dMax, mMax} (from T. D. Noe)
%Y A085000 Cf. A088214, A088215, A088216, A088217, A088237.
%Y A085000 Sequence in context: A162677 A041767 A130557 this_sequence A126154 A001327 
               A159533
%Y A085000 Adjacent sequences: A084997 A084998 A084999 this_sequence A085001 A085002 
               A085003
%K A085000 nonn,nice,hard
%O A085000 1,2
%A A085000 Robert G. Wilson (rgwv(AT)rgwv.com), Jun 16 2003
%E A085000 a(4) from Marsac Laurent (jko(AT)rox0r.net), Sep 15 2003
%E A085000 a(6) from Hugo Pfoertner (hugo(AT)pfoertner.org), Sep 23 2003
%E A085000 Entry edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 22 2006, 
               to remove some erroneous entries. Further edits Nov 25 2006.
%E A085000 a(7) from Hugo Pfoertner (hugo(AT)pfoertner.org), Jan 22 2008