[seqfan] Dragon sign sequence
Benoit CLOITRE
benoit7848c at orange.fr
Sat Mar 19 10:04:05 CET 2011
Dear seqfans,
Philippe Torgue found an amusing result relating the Dragon sequence to Pi. Define the Dragon sign sequence as follows:
d(1)=1 and for n>=2 by d(2n-1)=(-1)^n and d(2n)=d(n) so that d(n)=(-1)^A014707(n).
Then he claimed sum(n>=1,d(n)/n)=Pi/2 and didn't give his proof but this result is easy to prove and there is a slight generalisation.
Let F(s)=sum_{n>=1}(-1)^{n-1}/(2n-1)^s and D(s)=sum_{n>=1}(-1)^{n-1}/n^s
then we have
(2^s-1)D(s)=2^sF(s).
Note there is already something for the Thue-Morse sequence and Pi. See formula (3) and (4) at:
http://mathworld.wolfram.com/Thue-MorseSequence.html
I wonder if someone was aware about this fact before Torgue's simple discovery.
BC
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