# [seqfan] A connection between the Thue-Morse sequence A010060 and A000123

Vladimir Reshetnikov v.reshetnikov at gmail.com
Fri Nov 11 00:12:41 CET 2016

```Dear SeqFans,

The Thue-Morse sequence [0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, ...]
appears in the OEIS as http://oeis.org/A010060. It can be constructed
starting with 0 and repeatedly appending the binary complement of already
constructed part. Also, A010060(n) can be thought of as the parity of the
number of 1s in the binary representation of n.

Consider a power series f(x) = Sum_{n>=0} (-1)^A010060(n) * x^n = 1 - x -
x^2 + x^3 - x^4 + x^5 + x^6 - x^7 - x^8 + ...

Here is a graph of the function represented by this power series (if you
cannot see the graph inline in this message, you should be able to find it
at http://s16.postimg.org/c2qiykfud/Thue_Morse.png).

​
Its reciprocal (multiplicative inverse) is 1/f(x) = 1 + x + 2*x^2 + 2*x^3 +
4*x^4 + 4*x^5 + 6*x^6 + 6*x^7 + 10*x^8 + 10*x^9 + ..., where the
coefficients seem to go in pairs: [1, 1, 2, 2, 4, 4, 6, 6, 10, 10, ...].
Taking a bisection, we get the sequence [1, 2, 4, 6, 10, 14, 20, 26, 36,
46, 60, 74, 94, 114, 140, 166, ...]. A lookup in the OEIS returns a
possible match http://oeis.org/A000123, whose name is "Number of binary
partitions: number of partitions of 2n into powers of 2", and which has a
few other combinatorial interpretations.

Is it indeed that sequence? If so, can we infer any interesting results
from this connection?

--
Thanks