# 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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