A175844 Parse the base-2 expansion of 1/n using the Ziv-Lempel encoding as described in A106182; sequence gives the eventual period of the differences of the sequence of lengths of the successive phrases.

1, 1, 4, 1, 16, 4, 9, 1, 36, 16, 100, 4, 144, 9, 16, 1, 64, 36, 324, 16, 36, 100, 121, 4, 400, 144, 324, 9

Offset

1,3

Comments

The Ziv-Lempel encoding scans the sequence from left to right and inserts a comma when the current phrase (since the last comma) is distinct from all previous phrases (between commas).

It appears that a(n) is just the square of the period of the base 2 expansion of 1/n. For example, if n=3 the sequence of terms in the base-2 expansion of 1/3 is {0,1,0,1,0,1,0,1,...}, of period 2, whereas a(3)=4=2^2.

Links

Table of n, a(n) for n=1..28.

Example

For n=3, the sequence of base-2 digits of 1/3 is {0,1,0,1,0,1,0,1,0,1,0,1,...}. The Ziv-Lempel encoding parses this into "phrases": {0}, {1}, {0,1}, {0,1,0}, {1,0}, {1,0,1}, {0,1,0,1}, {0,1,0,1,0}, {1,0,1,0}, {1,0,1,0,1}, {0,1,0,1,0,1}, ..., with lengths {1,1,2,3,2,3,4,5,4,5,6,7,6,7,8,9,8,9,10,11,...}. The differences are {0,1,1,-1,1,1,1,-1,1,1,1,-1,1,...} which quickly becomes periodic with period 4. Thus a(3)=4.

Crossrefs

Cf. A106182, A109337.

Sequence in context: A062780 A262616 A309074 * A351434 A167343 A094361

Adjacent sequences: A175841 A175842 A175843 * A175845 A175846 A175847

Keyword

nonn

Author

John W. Layman, Sep 24 2010

Status

approved