[math-fun] More Median Musings

[Richard Guy]

Perhaps not quite all.  6 & 8 should be swapped.
Then the two sequences are

1  2  6  14  35   84  204  ...
1  3  8  20  49  119  288  ...

which are A105635 & A048739.  Some further
comment could be added to each of these.
They are closely related to the Bhaskara
equation, and are really second order
recurrences with a phony root, 1, appended,
making them cubic.

But it's much more natural to look at

         A001109    A053141
         A001108    A001652
Best to all,   R.

On Sun, 16 Oct 2005, David Gale wrote:

> Neil, Unfortunately I don't have access at
> the moment to the OEIS, but here is 
> the whole story on the median sequence.
> I'll be interested to know whether 
> and if so how it qualifies for inclusion.
>
> The integer m in N={1,2,..,n} is a MEDIAN of n if 1+..+m=m+..+n.
> I write the pair (n,m) . Example: (1,1) is a median pair
>> The integer m' in N={1,2,..,n} is a SUBMEDIAN of n if 1+..+m'=(m'+1)+..+n.
> I write the pair (n,m'). Example: (3,2) is a submedian pair.
> One wants to know which n have medians and submedians. Here are the first 7 
> pairs, alternately median and submedian.
>
> (1,1) med.
> (3,2) submed.
> (6,8) med.
> (20,14) submed. etc.
> (49,35)
> (119,84)
> (288,204)
> ......
> Here is the fundemental recusion:
> If (n,m) is a median pair then(n+2m,m+n) is  a submedian pair.
> If (n,m') is a submedian pair them (n+2m'+1,n+m'+1.) is a median pair.
>
> C'est tout.
>
> David
>
>> David,  is this sequence:
>> For which n does the sequence (1,2,  . ,n) have a perfect median?
>> , in the OEIS?  If so that property should be noted there,
>> and if not the sequence should be added!
>> Similarly for your other sequence.  Can you send me the first few terms?
>> You mentioned a Pellian equation.
>> 
>> Thanks!
>> 
>> Neil
>
> At 08:57 AM 10/6/2005, you wrote:
>> David,  is this sequence:
>> For which n does the sequence (1,2,  . ,n) have a perfect median?
>> , in the OEIS?  If so that property should be noted there,
>> and if not the sequence should be added!
>> Similarly for your other sequence.  Can you send me the first few terms?
>> You mentioned a Pellian equation.
>> 
>> Thanks!
>> 
>> Neil
>>
>>  Neil J. A. Sloane
>>  AT&T Shannon Labs, Room C233,
>>  180 Park Avenue, Florham Park, NJ 07932-0971
>>  Email: njas at research.att.com
>>  Office: 973 360 8415; fax: 973 360 8178
>>  Home page: http://www.research.att.com/~njas/
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