[seqfan] Re: A090381 "more" matched by new A245870

[Neil Sloane]

Followup on Ron Hardin's sequence. I've merged A090381 and A245870 (into
the older A-number, as usual), and I created two
new entries for the A and B triangles that I mentioned, A245556 and A245557.

(Still have to do a bit of editing)


On Mon, Aug 4, 2014 at 3:56 PM, Neil Sloane <njasloane at gmail.com> wrote:

> PS
> One thing I said wasn't quite right. The correct statement is that Ron's
> sequence 1,6,19,36,61,... (n>=0) is given by
> a(2t) = 12t^2+6t+1, a(2t+1) = 12t^2+18t+6.
>
> This is proved by summing the rows of the B triangle
> that I mentioned.
>
> The formula for the B triangle is B(0,0)=1,
>
> B(n,k) = 3k (0 <= k <= n-1)
> B(n,k) = 12n-3k-3 (n <= k <= 2n-1),
> B(n,2n) = 3n+1
>
> and - though I haven't written out all the details -
> these follow from simple counting arguments.
> (For the B array, we have to look at the following kinds of triples:
> i,j,n (i<j<n),
> i,i,n (i<n)
> i,n,n (i<n)
> n,n,n,
> and look at the pair sums)
>
> Neil
>
>
>
> On Mon, Aug 4, 2014 at 2:38 PM, Neil Sloane <njasloane at gmail.com> wrote:
>
>> Some observations about Ron's question:
>>
>> Let A(n,k) denote the number of triples (u,v,w) with entries in the range
>> 0 to n which have some pair adding up to k;
>> let B(n,k) be the same but count only cases in which at least
>> one of u v w is actually equal to n. (The range of k for both A and B is
>> 0 to 2n)
>>
>> The initial values of the A and B triangles are (n>=0, 0 <=k<=2n):
>>
>> [1]
>> [4, 6, 4]
>> [7, 12, 19, 12, 7]
>> [10, 18, 28, 36, 28, 18, 10]
>> [13, 24, 37, 48, 61, 48, 37, 24, 13]
>> [16, 30, 46, 60, 76, 90, 76, 60, 46, 30, 16]
>> [19, 36, 55, 72, 91, 108, 127, 108, 91, 72, 55, 36, 19]
>> [22, 42, 64, 84, 106, 126, 148, 168, 148, 126, 106, 84, 64, 42, 22]
>>
>> and B is
>>
>> [1]
>> [3, 6, 4]
>> [3, 6, 15, 12, 7]
>> [3, 6, 9, 24, 21, 18, 10]
>> [3, 6, 9, 12, 33, 30, 27, 24, 13]
>> [3, 6, 9, 12, 15, 42, 39, 36, 33, 30, 16]
>> [3, 6, 9, 12, 15, 18, 51, 48, 45, 42, 39, 36, 19]
>> [3, 6, 9, 12, 15, 18, 21, 60, 57, 54, 51, 48, 45, 42, 22]
>>
>> The rows of A are the partial sums of the rows of B, so it is
>> enough to explain B.
>>
>> The central spine of B is (all that follows is empirical
>> but should not be hard to prove) 1 followed by 9n-3.:
>> 1,6,15,24,33, ... = A017233
>>
>> The first half of each row of B is B(n,k) = 3k. The diagonals of the
>> second half are 3n+1, 6n, 6n+3, 6n+6, 6n+9, ... So B is simple,
>> and therefore so is A.
>>
>> Ron is asking about the central spine of A,
>> 1, 6, 19, 36, 61, ...
>> which incidentally is a mixture of two quadratics,
>> 1 19 61 ,,, which is 12t^2+6t+1,
>> and 6 36 90 168 ..., which is 6 times A017233.
>>
>> Neil
>>
>>
>>
>>
>> On Mon, Aug 4, 2014 at 1:15 PM, Ron Hardin <rhhardin at att.net> wrote:
>>
>>>
>>> A245870
>>>  Number of length 1+2 0..n arrays with some pair in every consecutive
>>> three terms totalling exactly n
>>>
>>>
>>>  6, 19, 36, 61, 90, 127, 168, 217, 270, 331, 396, 469, 546, 631, 720,
>>> 817, 918, 1027, ...
>>>
>>> matches
>>>
>>>
>>> A090381
>>>  Degree of toric ideal associated with path with n nodes.
>>>
>>>  6, 19, 36, 61, 90, 127, 168, 217, 270
>>>
>>> which asks for "more," if anybody can prove they're the same.
>>>
>>> (yahoo webmail is putting strange marks in my cut and paste work.  Who
>>> knows if this means formatting will totally fail.  WYSIWYG is of the past.)
>>>
>>>
>>> rhhardin at mindspring.com
>>> rhhardin at att.net (either)
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>
>>
>>
>> --
>> Dear Friends, I have now retired from AT&T. New coordinates:
>>
>> Neil J. A. Sloane, President, OEIS Foundation
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>>
>>
>
>
> --
> Dear Friends, I have now retired from AT&T. New coordinates:
>
> Neil J. A. Sloane, President, OEIS Foundation
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>


-- 
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com