[seqfan] Re: Dragon sign sequence
allouche at math.jussieu.fr
allouche at math.jussieu.fr
Sat Mar 19 10:11:15 CET 2011
Dear Benoit
Yes the formula with pi/2 was discovered by F. von Haeseler.
The more general case is exercise 27, page 205--206 in
the book by J. Shallit and myself:
Automatic Sequences: Theory, Applications, Generalizations
ambridge, 2003.
best
jean-paul
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Benoit CLOITRE <benoit7848c at orange.fr> a écrit :
> 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
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>
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