An Observation on Wichmann Rulers
Or: Are optimal rulers of Wichmann type?

B. Wichmann. A note on restricted difference bases.
J. London Math.Soc.38,1962,465-466

Note in the following table the red entries. They are identical to the entries in the sequence A004137 in the On-Line Encyclopedia of Integer Sequences for n=14..21, (length of optimal rulers with n-1 segments). This observation raises the question, whether or not the Sequence A004137 also continues like the sequence below (labeled L).

I conjectured in an informal context in a newsgroup (de.sci.mathematik) this being the case. However, in view of the small basis of evidence at the current state, this conjecture should preferably be regarded as a challenge for research. In any case it is an interesting observation and an investigation of the relation between optimal rulers and Wichmann rulers appears to be a worthwhile endeavor.

For clearness I state this 'conjecture':

In the set of optimal rulers with S segments where S>12
there exists at least one ruler of Wichmann type,
i. e. a ruler and integers r>0 and s>0 exist
such that the difference representation of the ruler has the form
1^r, r+1, (2r+1)^r, (4r+3)^s, (2r+2)^(r+1), 1^r.

A sharper form of the conjecture is: All optimal rulers with more than 13 segments are either  Wichmann rulers or the mirror images of Wichmann rulers.

[S:2  L:3  ] W(0,0 ) 12
[S:3  L:6  ] W(0,1 ) 132
[S:4  L:9  ] W(0,2 ) 1332
[S:8  L:29 ] W(1,2 ) 12377441
[S:9  L:36 ] W(1,3 ) 123777441
[S:10 L:43 ] W(1,4 ) 1237777441
[S:11 L:50 ] W(1,5 ) 12377777441
[S:13 L:68 ] W(2,3 ) 11355;;;66611
[S:14 L:79 ] W(2,4 ) 11355;;;;66611
[S:15 L:90 ] W(2,5 ) 11355;;;;;66611
[S:16 L:101] W(2,6 ) 11355;;;;;;66611
[S:17 L:112] W(2,7 ) 11355;;;;;;;66611
[S:18 L:123] W(2,8 ) 11355;;;;;;;;66611
[S:18 L:123] W(3,4 ) 1114777????8888111
[S:19 L:138] W(3,5 ) 1114777?????8888111
[S:20 L:153] W(3,6 ) 1114777??????8888111
[S:21 L:168] W(3,7 ) 1114777???????8888111
[S:22 L:183] W(3,8 ) 1114777????????8888111
[S:23 L:198] W(3,9 ) 1114777?????????8888111
[S:24 L:213] W(3,10) 1114777??????????8888111
[S:24 L:213] W(4,6 ) 111159999CCCCCC:::::1111
[S:25 L:232] W(4,7 ) 111159999CCCCCCC:::::1111
[S:26 L:251] W(4,8 ) 111159999CCCCCCCC:::::1111
[S:27 L:270] W(4,9 ) 111159999CCCCCCCCC:::::1111
[S:28 L:289] W(4,10) 111159999CCCCCCCCCC:::::1111
[S:29 L:308] W(4,11) 111159999CCCCCCCCCCC:::::1111
[S:30 L:327] W(4,12) 111159999CCCCCCCCCCCC:::::1111
[S:30 L:327] W(5,8 ) 111116;;;;;GGGGGGGG<<<<<<11111
[S:31 L:350] W(5,9 ) 111116;;;;;GGGGGGGGG<<<<<<11111
[S:32 L:373] W(5,10) 111116;;;;;GGGGGGGGGG<<<<<<11111
[S:33 L:396] W(5,11) 111116;;;;;GGGGGGGGGGG<<<<<<11111
[S:34 L:419] W(5,12) 111116;;;;;GGGGGGGGGGGG<<<<<<11111
[S:35 L:442] W(5,13) 111116;;;;;GGGGGGGGGGGGG<<<<<<11111
[S:36 L:465] W(5,14) 111116;;;;;GGGGGGGGGGGGGG<<<<<<11111
[S:36 L:465] W(6,10) 1111117======KKKKKKKKKK>>>>>>>111111
[S:37 L:492] W(6,11) 1111117======KKKKKKKKKKK>>>>>>>111111
[S:38 L:519] W(6,12) 1111117======KKKKKKKKKKKK>>>>>>>111111
[S:39 L:546] W(6,13) 1111117======KKKKKKKKKKKKK>>>>>>>111111
[S:40 L:573] W(6,14) 1111117======KKKKKKKKKKKKKK>>>>>>>111111
[S:41 L:600] W(6,15) 1111117======KKKKKKKKKKKKKKK>>>>>>>111111
[S:42 L:627] W(6,16) 1111117======KKKKKKKKKKKKKKKK>>>>>>>111111
[S:42 L:627] W(7,12) 11111118???????OOOOOOOOOOOO@@@@@@@@1111111
[S:43 L:658] W(7,13) 11111118???????OOOOOOOOOOOOO@@@@@@@@1111111
[S:44 L:689] W(7,14) 11111118???????OOOOOOOOOOOOOO@@@@@@@@1111111
[S:45 L:720] W(7,15) 11111118???????OOOOOOOOOOOOOOO@@@@@@@@1111111
[S:46 L:751] W(7,16) 11111118???????OOOOOOOOOOOOOOOO@@@@@@@@1111111
[S:47 L:782] W(7,17) 11111118???????OOOOOOOOOOOOOOOOO@@@@@@@@1111111
[S:48 L:813] W(7,18) 11111118???????OOOOOOOOOOOOOOOOOO@@@@@@@@1111111
[S:48 L:813] W(8,14) 111111119AAAAAAAASSSSSSSSSSSSSSBBBBBBBBB11111111
[S:49 L:848] W(8,15) 111111119AAAAAAAASSSSSSSSSSSSSSSBBBBBBBBB11111111

How to read the table:

S = number of segments, S = 4r+s+2.
L = length, L = 4r(r+s+2)+3(s+1). 
r and s are the Wichmann-parameters of rulers of type
W(r,s) = [1^r,r+1,(2r+1)^r,(4r+3)^s,(2r+2)^(r+1),1^r]

The rulers are represented as differences and coded as ASCII-strings. To get the numerical value subtract 48 from the ASCII-value. (Maple support file.)

Optimal rulers in standard representation here.