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%I #11 Mar 11 2024 13:35:48
%S 0,0,0,4,0,4,8,8,8,8,12,16,24,16,12,16,24,44,44,24,16,20,36,64,80,64,
%T 36,20,24,48,92,116,116,92,48,24,28,60,124,164,168,164,124,60,28,32,
%U 72,156,220,240,240,220,156,72,32,36,84,192,276,324,344,324,276,192,84,36,40,96
%N T(n,k) = number of 1:3:sqrt(10) proportioned triangles on a (n+1) X (k+1) grid.
%H R. H. Hardin, <a href="/A190103/b190103.txt">Table of n, a(n) for n = 1..9786</a>
%F Empirical for columns:
%F k=1: a(n) = 4*n - 8 for n>1
%F k=2: a(n) = 12*n - 36 for n>4
%F k=3: a(n) = 40*n - 168 for n>8
%F k=4: a(n) = 80*n - 400 for n>11
%F k=5: a(n) = 132*n - 768 for n>14
%F k=6: a(n) = 224*n - 1588 for n>18
%F k=7: a(n) = 336*n - 2712 for n>21
%F k=8: a(n) = 468*n - 4200 for n>24
%F k=9: a(n) = 660*n - 6732 for n>28
%F k=10: a(n) = 880*n - 9884 for n>31
%F k=11: a(n) = 1128*n - 13740 for n>34
%F k=12: a(n) = 1456*n - 19476 for n>38
%F k=13: a(n) = 1820*n - 26264 for n>41
%F k=14: a(n) = 2220*n - 34208 for n>44
%e Table starts
%e ..0..0...4...8..12..16...20...24...28...32...36...40...44...48...52...56...60
%e ..0..0...8..16..24..36...48...60...72...84...96..108..120..132..144..156..168
%e ..4..8..24..44..64..92..124..156..192..232..272..312..352..392..432..472..512
%e ..8.16..44..80.116.164..220..276..340..412..484..560..640..720..800..880..960
%e .12.24..64.116.168.240..324..408..504..612..720..836..960.1084.1212.1344.1476
%e .16.36..92.164.240.344..464..588..732..892.1056.1232.1420.1612.1812.2020.2232
%e .20.48.124.220.324.464..624..792..988.1204.1428.1672.1932.2200.2480.2772.3072
%e .24.60.156.276.408.588..792.1008.1260.1536.1824.2140.2476.2824.3192.3576.3972
%e .28.72.192.340.504.732..988.1260.1584.1936.2304.2712.3144.3592.4072.4572.5088
%e .32.84.232.412.612.892.1204.1536.1936.2368.2820.3328.3864.4420.5020.5644.6288
%e Some solutions for n=7 k=5
%e ..0..3....3..3....0..0....2..1....3..5....1..1....2..4....7..0....4..2....0..1
%e ..0..1....3..2....0..2....2..0....0..2....1..3....2..1....1..0....1..2....0..2
%e ..6..3....6..3....6..0....5..1....4..4....7..1....3..4....7..2....4..1....3..1
%o (PARI) T(n,k)=2*sum(i=0, n\3, sum(j=0, k\3, ((i!=0) + (j!=0))* (max(0, n+1 - max(3*i,j)) * max(0, k+1 - (3*j+i)) + max(0, n+1 - (3*i+j)) * max(0, k+1 - max(3*j, i)) ))) \\ _Andrew Howroyd_, Mar 11 2024
%Y Diagonal is A190102.
%Y Cf. A189885.
%K nonn,tabl
%O 1,4
%A _R. H. Hardin_, May 04 2011
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