[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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