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

[Neil Sloane]

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