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<div class=Section1>
<p class=MsoNormal><span style='font-family:"Courier New"'>I wrote a
brute-force program to compute A094913, which I ran over the weekend. The
results are as follows:<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>a(n) = maximum
number of distinct substrings of a binary string of length n.<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>b(n) = number of
length-n binary strings with a(n) distinct substrings.<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>c(n) = decimal value
of d(n) interpreted as a binary number.<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>d(n) = lexically
first length-n binary string with a(n) distinct substrings.<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
n a(n) b(n) c(n)<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
0 1
1 null<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
1 2
2 0<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
2 4
2 01<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
3 6
6 001<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
4 9
8 0010<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
5 13 4
00110<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
6 17 18
000110<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
7 22 38
0001011<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
8 28 48
00010110<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
9 35 40
000101100<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
10 43 16
0001011100<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
11 51 80
00001011100<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
12 60 210
000010011101<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
13 70 402
0000100110111<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
14 81 644
00001001101110<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
15 93 852
000010011010111<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
16 106 928
0000100110101110<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
17 120 912
00001001101011100<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
18 135 704
000010011010111000<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
19 151 256
0000100110101111000<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
20 167 1344
00000100110101111000<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
21 184 3944
000001000110101111001<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
22 202 9276
0000010001100101111010<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
23 221 19448 00000100011001010111101<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
24 241 37090 000001000110010101111010<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
25 262 65602
0000010001100101001111011<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
26 284 107388 00000100011001010011101111<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
27 307 160760 000001000110010100111011110<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>
28 331 220200 0000010001100101001110101111<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>Where both are
defined, a(n) = A094913(n)+1. I would suggest replacing A094913 with a(n),
since a(n) treats the empty string as a possible substring.<o:p></o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'><o:p> </o:p></span></p>
<p class=MsoNormal><span style='font-family:"Courier New"'>The new elements
a(19) through a(28) support Jovovic’s conjecture that a(n) = A006697(n).<o:p></o:p></span></p>
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