[seqfan] Re: Big Numbers in the Champernowne Continued Fraction Expansion

Hans Havermann pxp at rogers.com
Wed Mar 31 06:52:59 CEST 2010

Alonso Del Arte:

> Does A038705(0) = 1 sound correct to you?

Why does even this simple question seem so complicated to me?

The simple continued fraction expansion for Champernowne is 0, 8, 9,  
1, 149083, ... (A030167, offset-zero). The positions of the  
incrementally largest terms therein are 2, 3, 5, ... (my A038705,  
offset-one). Why didn't I start with 1? It was unclear to me ten years  
ago that I needed to. Perhaps the word "incrementally" suggested to my  
mind that the initial term be ignored. More likely I used a template,  
such as Eric Weisstein's 2, 5, ... (A033263, offset-one), positions of  
the incrementally largest terms in 0, 1, 1, 1, 27, ... (A033260,  
offset-one: it should be offset-zero, I think) as a guide.

Eric's submission of 0, 1, 2, 4, ... (A143533) differs in substance  
from his own many-years-earlier submission (above) in three distinct  

#1. He considers and prepends that first term.
#2. He uses index numbers instead of position numbers.
#3. He gives it an offset-zero instead of offset-one.

With regard to #1, is there now any consistency in the database with  
such "position" sequences? Perhaps it depends on the wording in the  
definition. Should there be consistency? Will one retrofit sequences  
to conform to an established norm?

With regard to #2, I actually prefer index numbers. I used them in my http://chesswanks.com/pxp/cfcd.html 
  wherein I suggest explicitly that position = index + 1, but of  
course one's linguistic mileage will vary. At any rate, I submitted  
A038705 as position numbers instead of index numbers back then  
because, at the time, that's how I saw it being done. The previous  
question about retrofitting applies.

With regard to #3, I'm not at all sure but I think it's offset-one  
irrespective of what one does in either #1 or #2.

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