[seqfan] Re: An equivalence for integer sequences (with more questions than answers)
Paolo Lava
ppl at spl.at
Fri Feb 27 09:56:10 CET 2009
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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