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

[Ron Hardin]

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)