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

[Paolo Lava]

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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