# [seqfan] Re: Right Triangle Count on an nXk grid - same effect as rhombuses

Ron Hardin rhhardin at att.net
Mon Apr 25 12:58:39 CEST 2011

```Formula guess-the-coefficients list extended to 14
k=1 Empirical: a(n) = 2*n^2 + 4*n - 2
k=2 Empirical: a(n) = 6*n^2 + 26*n - 42 for n>3
k=3 Empirical: a(n) = 12*n^2 + 88*n - 240 for n>8
k=4 Empirical: a(n) = 20*n^2 + 228*n - 930 for n>15
k=5 Empirical: a(n) = 30*n^2 + 468*n - 2478 for n>24
k=6 Empirical: a(n) = 42*n^2 + 886*n - 6080 for n>35
k=7 Empirical: a(n) = 56*n^2 + 1480*n - 12216 for n>48
k=8 Empirical: a(n) = 72*n^2 + 2344*n - 23112 for n>63
k=9 Empirical: a(n) = 90*n^2 + 3516*n - 40434 for n>80
k=10 Empirical: a(n) = 110*n^2 + 5090*n - 67626 for n>99
k=11 Empirical: a(n) = 132*n^2 + 7016*n - 105016 for n>120
k=12 Empirical: a(n) = 156*n^2 + 9564*n - 162094 for n>143
k=13 Empirical: a(n) = 182*n^2 + 12572*n - 236518 for n>168
k=14 Empirical: a(n) = 210*n^2 + 16230*n - 337676 for n>195

rhhardin at mindspring.com
rhhardin at att.net (either)

----- Original Message ----
> From: Ron Hardin <rhhardin at att.net>
> To: seqfan at list.seqfan.eu
> Sent: Sun, April 24, 2011 9:26:46 PM
> Subject: [seqfan] Right Triangle Count on an nXk grid - same effect as
>rhombuses
>
> The same thing happens for right triangles on a nXk grid as for rhombuses - the
>
> parabolic settling into a recurrence independent of k for large  n.
>
> T(n,k)=Number of right triangles on a (n+1)X(k+1) grid
>
> Table  starts
> ...4..14...28...46...68...94...124...158...196...238...284...334...388...446
> ..14..44...94..158..238..330...434...550...678...818...970..1134..1310..1498
> ..28..94..200..342..524..732...972..1236..1524..1840..2180..2544..2932..3344
> ..46.158..342..596..926.1308..1754..2250..2794..3390..4026..4702..5426..6190
> ..68.238..524..926.1444.2060..2784..3596..4492..5470..6516..7630..8820.10070
> ..94.330..732.1308.2060.2960..4032..5250..6604..8082..9684.11388.13220.15144
> .124.434..972.1754.2784.4032..5520..7224..9128.11218.13500.15938.18568.21328
> .158.550.1236.2250.3596.5250..7224..9496.12044.14860.17948.21266.24852.28634
> .196.678.1524.2794.4492.6604..9128.12044.15332.18990.23012.27354.32052.37032
> .238.818.1840.3390.5470.8082.11218.14860.18990.23596.28678.34190.40166.46522
>
> k=1  Empirical: a(n) = 2*n^2 + 4*n - 2
> k=2 Empirical: a(n) = 6*n^2 + 26*n - 42  for n>3
> k=3 Empirical: a(n) = 12*n^2 + 88*n - 240 for n>8
> k=4  Empirical: a(n) = 20*n^2 + 228*n - 930 for n>15
> k=5 Empirical: a(n) =  30*n^2 + 468*n - 2478 for n>24
> k=6 Empirical: a(n) = 42*n^2 + 886*n - 6080  for n>35
> k=7 Empirical: a(n) = 56*n^2 + 1480*n - 12216 for n>48
> k=8  Empirical: a(n) = 72*n^2 + 2344*n - 23112 for n>63
>
> obviously  n>k^2-1 is where the parabola kicks in.
>
> Diagonal is http://oeis.org/A077435
>
> There's a game of guess the coefficients  open here.
>
>
>
> rhhardin at mindspring.com
> rhhardin at att.net (either)
>
>
> _______________________________________________
>
> Seqfan  Mailing list - http://list.seqfan.eu/
>

```