Boustrophedon transformation of constants

[Mark Hudson]

Hi,

I was wondering if anyone thinks, or has thought, that applying the 
Boustrophedon transform to the continued fraction of some constants could be 
interesting?

What I mean is, that we take for instance Pi, and find its continued 
fraction:

[3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 
1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1,...]

Then, we apply the Boustrophedon transform to this sequence. This, in my 
calculation, leads to:

[3,10,32,73,457,1994,6407,29489,148253,852592,5420543,37975111,290066507,2400720769,21396506651,204322668174,2081209926313,22523982873141,258105780607144,...]

where I've left some of the larger terms off the end.

Now, this result could be interpreted as the elements of a continued 
fraction. The number that they lead to is:

3.0996886064030483425267288917220886641288797602530...

where I've actually used the first 200 continued fraction terms of Pi to 
calculate this in Maple.

I'd like to call this "the Boustrophedon transform of Pi".

I can't see any particular use for this, but it seemed interesting to me...

Mark.









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