Wrong new terms in A085000

[Hugo Pfoertner]

Seqfans, Neil,

the sequence A085000 "Maximal determinant of an n X n matrix using the
integers 1 to n^2."

that had previously the terms a(1)...a(6)
1,10,412,40800,6839492,1865999570,

has recently been extendend as follows:

762074188050,440477621828418, 345964972957674150

The comment says:
a(7) - a(9) were found by Richard Mueller
(richard.mueller(AT)informatik.hu-berlin.de) and Johannes Dewender
(johannes.dewender(AT)informatik.hu-berlin.de) in cooperation, Oct 22
2006 
Entry updated by Richard Mueller  Oct 22 2006

Unfortunately bigger counterexamples exist for all 3 terms:

d7:=det mat
((28, 9,34,41, 1,40,22),
 (47,25, 7,36,35,16, 8),
 (3, 27, 6,39,31,26,43),
 (38,21,32,14,18, 4,48),
 (19, 5,37,13,49,33,20),
 (29,42,15, 2,17,45,24),
 (11,46,44,30,23,12,10));
 
 Reduce: d7 := 762139309293

d8:= det mat
((13, 53, 40, 25,  2, 45, 58, 23),
 (21, 22,  8, 54, 24, 56, 18, 57),
 (60, 11, 36, 44, 12,  7, 49, 41),
 (15, 19, 48,  4, 52, 26, 38, 59),
 (20, 62, 34, 51, 42,  1, 17, 33),
 (37, 29,  3, 31, 61, 35, 55, 10),
 (30, 14, 63, 46, 39, 47, 16,  5),
 (64, 50, 28,  6, 27, 43,  9, 32));

Reduce: d8 := 440857916120379

d9:= det mat
((40, 48, 72, 26,  6, 50, 39, 78, 10),
 (23, 77, 42,  7, 37, 41, 74, 12, 56),
 (52, 47,  8, 54,  5, 64, 19, 44, 75),
 (59, 25, 13, 43, 58, 76, 63, 30,  3),
 (79, 36, 71, 60, 31, 18, 34,  2, 38),
 (32, 81, 22, 61, 67, 15, 17, 53, 21),
 (20, 24, 66, 27, 73, 68,  4, 33, 55),
 ( 1, 14, 45, 80, 35, 29, 69, 46, 49),
 (65, 16, 28, 11, 57,  9, 51, 70, 62));

Reduce: d9 := 346202123011922653

(Thanks to Rainer Rosenthal for X-checking the determinants with Maple).

None of the counterexamples is known to be the biggest possible
solution. The best value I've found in 2 CPU years for
a(7)=762140212575, but currently I don't have access to the computer
where I've found this solution and the corresponding arrangement of
matrix elements.

My suggestion is to remove the 3 new terms and the new Mathematica
program and to restore the previous state of the sequence. The
determination of a(8) and a(9) is currently beyond our available
computational capabilities. I will try to confirm a(7) and to find
improved lower bounds for a(8) and a(9) by running the slightly improved
program given at the link for some months on a cluster of Intel Itanium
2 CPUs.

In the meantime I suggest to add a comment:

Known lower bounds for a(7)...a(9) are
762140212575, 440857916120379, 346202123011922653,

Hugo Pfoertner