# [seqfan] Re: No isosceles triangles in a square grid, continued

Neil Sloane njasloane at gmail.com
Wed Apr 27 14:43:13 CEST 2016

```Your array isn't in the OEIS yet, I think, so could you submit it?

Best regards
Neil

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.
Email: njasloane at gmail.com

On Wed, Apr 27, 2016 at 8:12 AM, Ron Hardin <rhhardin at att.net> wrote:

> Okay, good.  I should drop the "right triangle" constraint.  For the right
> n+k-1 case i have so far
> T(n,k)=Number of nXk 0..1 arrays with exactly n+k-1 having value 1 and no
> three 1s forming an isosceles right triangle
>
> Table starts
> .1.1...1....1.....1....1....1....1.....1.....1....1....1.1.1
> .1.0...1....0.....1....0....1....0.....1.....0....1....0.1..
> .1.1...0....1.....9....6...16...66....95...177..493.1153....
> .1.0...1....0.....8....8...71..212...731..1840.5953.........
> .1.1...9....8....12...58..367.1952.10854.28952..............
> .1.0...6....8....58..192..838.4968.37436....................
> .1.1..16...71...367..838.4892...............................
> .1.0..66..212..1952.4968....................................
> .1.1..95..731.10854.........................................
> .1.0.177.1840...............................................
> .1.1.493....................................................
> .1.0........................................................
> .1..........................................................
>
> Empirical for column k:
> k=1: a(n) = a(n-1)
> k=2: a(n) = a(n-2)
> k=3: a(n) = 3*a(n-1) -3*a(n-2) +4*a(n-3) -6*a(n-5) +3*a(n-6) -6*a(n-7)
> -16*a(n-8) +44*a(n-9) -14*a(n-10) -42*a(n-11) +65*a(n-12) -56*a(n-13)
> +52*a(n-14) -8*a(n-15) -65*a(n-16) +63*a(n-17) -43*a(n-18) +64*a(n-19)
> -6*a(n-20) -119*a(n-21) +122*a(n-22) +42*a(n-23) -93*a(n-24) +16*a(n-25)
> +20*a(n-26) -6*a(n-27) -8*a(n-28) -2*a(n-30) -2*a(n-31) +4*a(n-32)
> probably the last recurrence within my reach.
>  rhhardin at mindspring.com rhhardin at att.net (either)
>
>
>       From: Neil Sloane <njasloane at gmail.com>
>  To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>  Sent: Wednesday, April 27, 2016 7:07 AM
>  Subject: [seqfan] Re: No isosceles triangles in a square grid, continued
>
> Ron, There are two separate problems being discussed:
>
> No right isosceles triangles:
> =====================
> NORIT(m,n) = max no of 1's in mXn array so that no 3 form a right isosceles
> triangle
> - this array is not yet in OEIS (Because I don't know the initial values -
> but I would like to)
>
> A271906(n) = main diagonal of that array = max no of 1's in nXn array so
> that no 3 form a right isosceles triangle. It is known that this grows like
> O(n^1.99999) but not as fast as O(n^2)  (!)  (paper in preparation, with
> Warren Smith, part of a general attack on problems of this class)
>
> No isosceles triangles:
> =================
> A271914(m,n) = max no of 1's in mXn array so that no 3 form an isosceles
> triangle, and
> I conjecture A271907(m,n) <= m+n-1. The main diagonal is conjectured to be
> = 2n-2 for n>1.
>
> A271907(n) = main diagonal of that array = max no of 1's in nXn array so
> that no 3 form an isosceles triangle,
> which I conjecture to equal 2n-2 for n>1.
>
> [The comment you quoted about "For (m,n) with 10 >= m >= n >= 2, the only
> exceptions [to m+n-2] are..."
> was referring to A271914(m,n). There are in fact infinitely
> many exceptions, thanks to Rob's work. But (I conjecture) all values are <=
> m+n-1.]
>
> Best regards
> Neil
>
> >
> >
>
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```