[seqfan] Repeated largest or second largest factor in term ratio

[Ron Hardin]

The ratio of terms of the following

1,2,4,8,32,192,1728,25920,518400,14515200,609638400,36578304000,
3292047360000,444426393600000,86663146752000000,24785659971072000000,
10360405867908096000000,6299126767688122368000000
Number of nXn symmetric binary matrices with each 1 adjacent to exactly 1 
antidiagonally neighboring 1
Some solutions for n=5
..0..0..0..0..0....0..1..0..0..0....0..1..0..0..1....0..1..0..0..0
..0..0..1..0..0....1..0..1..0..0....1..0..1..1..0....1..0..1..0..0
..0..1..0..1..0....0..1..0..1..0....0..1..0..1..0....0..1..0..1..0
..0..0..1..0..0....0..0..1..0..1....0..1..1..0..1....0..0..1..0..0
..0..0..0..0..0....0..0..0..1..0....1..0..0..1..0....0..0..0..0..0

has a peculiar property (table is a(n+1)/a(n) followed by its factors)

2: 2
2: 2
2: 2
4: 2 2
6: 2 3
9: 3 3
15: 3 5
20: 2 2 5
28: 2 2 7
42: 2 3 7
60: 2 2 3 5
90: 2 3 3 5
135: 3 3 3 5
195: 3 5 13
286: 2 11 13
418: 2 11 19
608: 2 2 2 2 2 19
896: 2 2 2 2 2 2 2 7
1316: 2 2 7 47
1927: 41 47
2829: 3 23 41
4140: 2 2 3 3 5 23
6060: 2 2 3 5 101
8888: 2 2 2 11 101
13024: 2 2 2 2 2 11 37
19092: 2 2 3 37 43
27993: 3 7 31 43
41013: 3 3 3 7 7 31
60102: 2 3 3 3 3 7 53
88086: 2 3 53 277
129082: 2 233 277
189196: 2 2 7 29 233
277298: 2 7 29 683
406385: 5 7 17 683
595595: 5 7 7 11 13 17
872872: 2 2 2 7 11 13 109
1279224: 2 2 2 3 3 109 163
1874826: 2 3 3 3 3 71 163
2747700: 2 2 3 3 5 5 43 71
4026950: 2 5 5 43 1873
5901823: 23 137 1873
8649495: 3 3 5 23 61 137
12676410: 2 3 3 5 61 2309
18578214: 2 3 3 3 149 2309

Start at the end, take the repeated factor (2309).  The largest other factor 
(61) will be in the preceding line.

Thus 2309 61 137 1873 43 71 163 109 13 17 683 29 233 277 53 7 31 43  37 11 101 5 
23 41 47 7 2 19 11 13 5 3 5 3 7 2 5 3 3 2

The series for diagonal instead of antidiagonal adjacency has an empirical 
formula because the ratio of its terms is A000930
a(n) = product(sum(binomial(k+1-2*i,i), i=0..floor((k+1)/3)), k=1..n)


 rhhardin at mindspring.com
rhhardin at att.net (either)