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

[Neil Sloane]

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