Fibs

[John Conway]

On Sun, 23 Jan 2000, N. J. A. Sloane wrote:

> Responding to john conway's comments:
> 
> If you enter
> 1 1 2 3 5 8 13 21 34 55 89 144 233 377
> the Fibonacci numbers are the 4th matching sequence

   You think this is good? I think it's absolutely terrible.
My guess is that the three before it are junk.  The Fibonacci 
sequence should be the first to come up after only half-a-dozen
terms.   

> There's now a button whereby you can control how
> many matching sequences you want to see (just as with Math Review
> lookups)

> I agree there are too many dull sequences in the table,
> and i hope readers will take your comments to heart.

   It seems to me that what you really need is some kind of
indexing of the sequences by "likeliness", and that providing
this is something that should be thought about very hard.  Even
a universal acceptance policy could be made to work if some
such indexing were provided.

   I've been thinking about it, and it seems to me that one get
something pretty good just by registering hits on the sequences.
It would need to be supplemented, though, because on the one
hand there are very important sequences that won't get hit because 
they're too simple, and others perhaps because they are too subtle
for the majority of users.

   One valuable way to supplement the hit count is to award weight
to sequences that have been discussed in published papers.  Another
good one is presence in the HIS &/or EIS.  Another is your own good
judgement, though that, of course, would be time-consuming.

   I strongly suggest that you think immediately about adding another
"interest" line, which would be modified automatically with time, and
also be manually adjustable.  Level 0 would mean that the only interest
there'd been in this sequence was by its contributor, although perhaps
the system might automatically evaluate contributors as well! 

> Here are a couple you will be glad i did not reject, taken just from
> the last couple of days' arrivals:
> 
> %I A052109
> %S A052109 1,1,2,5,12,30,73,178,434,1058,2580,6291,15341,37408,91217,222427,
> %T A052109 542374
> %N A052109 a(1)=1, a(n)=a(n-a(1))+a(n-a(2))+a(n-a(3))+....a(n-a(n-1)) for n>1, with convention that a(i)=0 for i<=0.
> %e A052109 a(5)=a(4)+a(4)+a(3)=5+5+2=12.
> %O A052109 1,3
> %K A052109 easy,nonn,nice,more
> %A A052109 Robert Lozyniak (rampshot at usa.net), Jan 20 2000
> 
> %I A052130
> %S A052130 1,2,7,15,37,84,187,421,914,2001,4283,9184
> %N A052130 For m very large, a(n) = number of numbers between 1 and 2^m with m-n prime factors (counted with multiplicity).
> %e A052130 Between 1 and 2^m there is just one number with m prime factor, namely 2^m, so a(0) = 1. There are 2 numbers with m-1 prime factors (2^(m-1) and 3*2^(m-2)), so a(1) = 2.
> %O A052130 0,2
> %K A052130 nonn,nice,more
> %A A052130 Bernd-Rainer Lauber (br.lauber at surf1.de), Jan 21 2000
> 
> Cute, don't you agree?

  Yes, I like these ones.

   Regards,  JHC