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

Author

John W. Layman, Sep 24 2010

Keywords

• nonn

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.

Examples

			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.