[seqfan] x^2=1/n sequences

bradklee at gmail.com bradklee at gmail.com
Sun Sep 27 19:43:39 CEST 2015


The following generalizes to arbitrary n, but for a first example take n=2.

Expanding the digits of 1/sqrt(2) in base-2, we get A004539. Taking the formal power series and squaring it, we obtain a new power series with coefficients:

0,1,0,2,2,1,4,1,4,2,2,3,0,2,2...

Not in OEIS. Clearly this sequence is equivalent via sqrt to A004539, but does it have any extra value?

We also have rewrite rules for changing this sequence into a proper binary sequence. The carry rules derive from shift-decrement equation

2x=1

Applying these rules we reduce the sequence to

0,1,0,2,2,1,4,1,4,2,2,3,0,2,2...
0,1,1,0,2,1,4,1,4,2,2,3,0,2,2...
0,1,1,1,0,1,4,1,4,2,2,3,0,2,2...
0,1,1,1,1,1,0,1,4,2,2,3,0,2,2...
...
0,1,1,1,1,1,1,1,1,1,1,1,1,1,0...

This is one binary representation of 1/2, and one of the easiest sequences in OEIS.

The new sequence is at once simple and complex. Maybe it will be easier to analyze than A004539.

Thanks,

Brad




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