Distinct Substrings In Binary n
Jonathan Post
jvospost3 at gmail.com
Mon Jun 23 19:46:51 CEST 2008
The distinct substrings (including null string) in binary
representation of trinary (A007089 Numbers in base 3):
n n(base 3) a(n) comment
0 0 2
1 1 2
2 2 2
3 10 4 = #{{},0,1,10}
4 11 3
5 12 4
6 20 4
7 21 4
8 22 3
9 100 6 = #{{},0,1,00,10,100}
10 101 6
11 102 7 = {{},0,1,2,10,02,102}
12 110 6
13 111 4 = {{},1,11,111}
14 112 6
15 120 7
16 121 6
17 122 6
18 200 6
19 201 7
20 202 6
21 210 7
22 211 6
23 212 6
24 220 6
25 221 6
26 222 4
27 1000 8 = {{},0,1,00,10,000,100,1000} etcetera
28 1001
29 1002
30 1010
31 1011
32 1012
33 1020
34 1021
35 1022
36 1100
37 1101
38 1102
39 1110
40 1111
41 1112
42 1120
43 1121
44 1122
45 1200
lq> From seqfan-owner at ext.jussieu.fr Mon Jun 23 19:22:52 2008
lq> Date: Mon, 23 Jun 2008 10:21:38 -0700 (PDT)
lq> From: Leroy Quet <q1qq2qqq3qqqq at yahoo.com>
lq> Subject: Distinct Substrings In Binary n
lq> To: seqfan at ext.jussieu.fr
lq> Cc: q1qq2qqq3qqqq at yahoo.com, qq-quet at mindspring.com
lq
lq> I am having a hard time believing that at least
lq> the first sequence listed below isn't in the
lq> database yet. Have I made an error with any of
lq> these sequences? I am waiting to submit them.
lq>
lq> (Including the empty substring does not return a
lq> result for the first sequence, all terms +1,
lq> either.)
lq>
lq> My terminology may be off.
lq>
lq> --
lq>
lq> a(n) = number of distinct (nonempty) substrings
lq> in the binary representation of n. (Leading
lq> zeroes are stripped.)
lq>
lq> Offset = 1.
lq>
lq> 1,3,2,5,5,5,3,7,8,7,...
lq>
lq> For example:
lq> The distinct substrings in binary representation
lq> (1010) of decimal 10:
lq> 0,1,10,01,101,010,1010.
The variant A122953 is in the OEIS. This sequence here continues
1,3,2,5,5,5,3,7,8,7,8,8,8,7,4,9,11,11,12,11,9,11,11,11,12,11,11,11,11,
9,5,11,14,15,16,14,15,16,16,15,15,11,14,16,14,15,14,14,16,16,16,16,14,14,15,
15,16,15,15,14,14,11,6,13,17,19,20,19,20,21,21,19,17,19,21,20,21,21,20,19,
Constructed with the Maple program
A := proc(n)
end:
for n from 1 to 80 do
od:
Richard
More information about the SeqFan
mailing list