%I #5 Mar 31 2012 12:36:19

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,8,16,8,0,0,0,0,12,28,

%T 28,12,0,0,0,0,16,40,48,40,16,0,0,0,0,20,56,68,68,56,20,0,0,0,0,24,72,

%U 96,96,96,72,24,0,0,0,0,28,88,124,136,136,124,88,28,0,0,0,0,32,104,152,180,192

%N T(n,k)=Number of 3:4:5 proportioned triangles on a (n+1)X(k+1) grid

%C Table starts

%C .0.0..0...0...0...0...0...0...0...0...0....0....0....0....0....0....0....0....0

%C .0.0..0...0...0...0...0...0...0...0...0....0....0....0....0....0....0....0....0

%C .0.0..0...4...8..12..16..20..24..28..32...36...40...44...48...52...56...60...64

%C .0.0..4..16..28..40..56..72..88.104.120..136..152..168..184..200..216..232..248

%C .0.0..8..28..48..68..96.124.152.180.208..236..264..292..320..348..376..404..432

%C .0.0.12..40..68..96.136.180.224.268.316..364..412..460..508..556..604..652..700

%C .0.0.16..56..96.136.192.256.320.384.456..528..600..672..744..816..888..960.1032

%C .0.0.20..72.124.180.256.344.432.524.628..732..836..944.1052.1160.1268.1376.1484

%C .0.0.24..88.152.224.320.432.544.664.800..940.1080.1228.1380.1532.1684.1840.1996

%C .0.0.28.104.180.268.384.524.664.816.988.1168.1348.1540.1740.1940.2140.2348.2556

%H R. H. Hardin, <a href="/A189973/b189973.txt">Table of n, a(n) for n = 1..10025</a>

%F Empirical for column k:

%F k=3: a(n) = 4*n - 12 for n>2

%F k=4: a(n) = 16*n - 56 for n>5

%F k=5: a(n) = 28*n - 100 for n>5

%F k=6: a(n) = 48*n - 212 for n>9

%F k=7: a(n) = 72*n - 336 for n>9

%F k=8: a(n) = 108*n - 568 for n>12

%F k=9: a(n) = 156*n - 968 for n>16

%F k=10: a(n) = 208*n - 1396 for n>16

%F k=11: a(n) = 264*n - 1844 for n>16

%F k=12: a(n) = 352*n - 2828 for n>23

%F k=13: a(n) = 444*n - 3856 for n>23

%F k=14: a(n) = 540*n - 4912 for n>23

%e Some solutions for n=7 k=5

%e ..3..3....1..5....2..0....2..4....7..1....6..4....7..5....7..1....4..3....2..3

%e ..3..0....1..1....2..4....2..0....3..1....3..4....4..5....4..1....0..3....2..0

%e ..7..3....4..5....5..0....5..4....7..4....6..0....7..1....7..5....4..0....6..3

%K nonn,tabl

%O 1,18

%A _R. H. Hardin_ May 03 2011