[seqfan] Re: An equivalence for integer sequences (with more questions than answers)

[David Wilson]

NOOOOOOOOOOOOOOOOOOOO........
Don't DOOOOOOOOOOOOO this.........

----- Original Message ----- 
From: "Paolo Lava" <ppl at spl.at>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Sent: Friday, February 27, 2009 3:56 AM
Subject: [seqfan] Re: An equivalence for integer sequences (with more 
questions than answers)


>
> Neil and seqfans,
>
> "More generally, for a strictly positive sequence with offset O,
> we can assign the value Sum_{n >= O} 1/a(n).
> If the sum doesn't converge, assign the value "NA" (Not Applicable)"
>
>
> To avoid “NA” we could consider the terms of the sequence as coefficients 
> of a simple continued fraction. For instance:
>
> a(n)                 N
> n             .6977746580… Positive integers
> 2*n             .4463899659… Even numbers
> 2*n-1             .7615941560… Odd numbers
> n^2             .8043185611… Squares
> 3/2+[(-1)^n]/2     .7320508076… Periodic seq 1,2,1,2,1,2….
> 3/2+[(-1)^(n+1)]/2  .3660254038 …       Periodic seq 2,1,2,1,2,1….
> (half of the previous)
> [(n+2) mod 3]+1     .6942541768… Periodic seq 1,2,3,1,2,3…
> [(n+1) mod 3]+1     .2706906326… Periodic seq 3,1,2,3,1,2…
> (n mod 3)+1     .4403946472… Periodic seq 2,3,1,2,3,1…
> n!             .6840959001… Factorials
> n!!             .6988047677… Double factorials
> n!-n             .3278711883… Factorials – integers
> n!+n             .4457779698… Fattorials + integers
> 4, 6, 8, 9, 10, 12  .2401934727… Composite numbers
> 2, 3, 5, 7, 11, 13  .4323320872… Prime numbers
>
> We could also consider a “distance” between two sequences as the absolute 
> value of the difference of their “N”s:
>
> a(n)=n     N1 = .6977746580…
> a(n)=n!    N2 = .6988047677…
>
> | N1-N2 | = .0136787579…
>
> Paolo
>
> --- njas at research.att.com wrote:
>
> From: "N. J. A. Sloane" <njas at research.att.com>
> To: seqfan at list.seqfan.eu
> Cc: njas at research.att.com
> Subject: [seqfan] Re: An equivalence for integer sequences (with more 
> questions than answers)
> Date: Thu, 26 Feb 2009 18:12:57 -0500
>
> Jaume made a very interesting comment, suggesting that we classify
> sequences { a(n): n >= 0 } by the value of Sum_{n >= 0} 1/a(n)
>
> More generally, for a strictly positive sequence with offset O,
> we can assign the value Sum_{n >= O} 1/a(n)
>
> If the sum doesn't converge, assign the value "NA" (Not Applicable)
>
> Even more generally, for a pair of sequences a(n), b(n),
> we could assign the value of Sum_{n >= O} a(n)/b(n).
>
> In fact the possibilities are endless for assigning numbers to
> sequences or pairs of sequences.  Lim_{n >= O} a(n)/b(n), for example.
>
> Neil
>
>
>
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