Distinct Substrings In Binary n
Leroy Quet
q1qq2qqq3qqqq at yahoo.com
Mon Jun 23 19:21:38 CEST 2008
I am having a hard time believing that at least
the first sequence listed below isn't in the
database yet. Have I made an error with any of
these sequences? I am waiting to submit them.
(Including the empty substring does not return a
result for the first sequence, all terms +1,
either.)
My terminology may be off.
--
a(n) = number of distinct (nonempty) substrings
in the binary representation of n. (Leading
zeroes are stripped.)
Offset = 1.
1,3,2,5,5,5,3,7,8,7,...
For example:
The distinct substrings in binary representation
(1010) of decimal 10:
0,1,10,01,101,010,1010.
So a(1) = 7.
--
b(n) = number of distinct substrings in the
binary representation of n that each occur
multiple times.
0,0,1,1,1,1,2,2,2,3,1,2,1,2,3
Example: The distinct substrings that occur
multiple times in 1010 are 0,1, and 10. So
a(10)=3.
--
c(n) = number of (not necessarily distinct)
substrings (with repetition) in the binary
representation of n that each occur multiple
times.
0,0,2,2,2,2,5,5,4,6,3,4,3,5,9
Example: 0,0,1,1,10,10 each occur multiple times
in 1010.
So a(10) = 6.
--
d(n) = number of distinct (nonempty) substrings
in the binary representation of n that each occur
exactly once.
1,3,1,4,4,4,1,5,6,4,7,6,7,5,1,
In 1010: 01, 101, 010, 1010 each occur exactly
once. So a(1) = 4.
---
a(n) = b(n) + d(n), obviously, it should be
noted.
Thanks,
Leroy Quet
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