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%I #11 Mar 11 2024 13:33:40
%S 0,0,0,0,0,0,4,0,0,4,8,8,0,8,8,12,16,12,12,16,12,16,24,24,32,24,24,16,
%T 20,32,36,56,56,36,32,20,24,44,48,80,96,80,48,44,24,28,56,68,104,136,
%U 136,104,68,56,28,32,68,88,140,176,192,176,140,88,68,32,36,80,108,180,232
%N T(n,k) = number of 1:4:sqrt(17) proportioned triangles on a (n+1) X (k+1) grid.
%H R. H. Hardin, <a href="/A189885/b189885.txt">Table of n, a(n) for n = 1..9232</a>
%F Empirical for columns:
%F k=1: a(n) = 4*n - 12 for n>2
%F k=2: a(n) = 12*n - 52 for n>6
%F k=3: a(n) = 24*n - 136 for n>10
%F k=4: a(n) = 60*n - 440 for n>15
%F k=5: a(n) = 108*n - 912 for n>19
%F k=6: a(n) = 168*n - 1600 for n>23
%F k=7: a(n) = 240*n - 2552 for n>27
%F k=8: a(n) = 360*n - 4428 for n>32
%F k=9: a(n) = 500*n - 6824 for n>36
%F k=10: a(n) = 660*n - 9820 for n>40
%F k=11: a(n) = 840*n - 13496 for n>44
%F k=12: a(n) = 1092*n - 19284 for n>49
%F k=13: a(n) = 1372*n - 26140 for n>53
%F k=14: a(n) = 1680*n - 34176 for n>57
%e Table starts
%e ..0..0...0...4...8..12..16..20...24...28...32...36...40...44...48...52...56
%e ..0..0...0...8..16..24..32..44...56...68...80...92..104..116..128..140..152
%e ..0..0...0..12..24..36..48..68...88..108..128..152..176..200..224..248..272
%e ..4..8..12..32..56..80.104.140..180..220..260..308..360..412..464..520..580
%e ..8.16..24..56..96.136.176.232..296..360..424..500..584..668..752..844..944
%e .12.24..36..80.136.192.248.328..420..512..604..712..832..952.1072.1204.1348
%e .16.32..48.104.176.248.320.424..544..664..784..928.1088.1248.1408.1584.1776
%e .20.44..68.140.232.328.424.560..716..876.1036.1228.1440.1656.1872.2112.2372
%e .24.56..88.180.296.420.544.716..912.1116.1320.1564.1832.2108.2384.2692.3024
%e .28.68.108.220.360.512.664.876.1116.1368.1620.1920.2248.2588.2928.3308.3716
%e Some solutions for n=6 k=4
%e ..2..0....0..2....6..3....0..4....2..3....1..4....5..3....0..4....1..3....0..0
%e ..2..4....0..1....2..3....0..3....2..4....0..4....1..3....0..0....1..2....0..4
%e ..3..0....4..2....6..2....4..4....6..3....1..0....5..4....1..4....5..3....1..0
%o (PARI) T(n,k)=2*sum(i=0, n\4, sum(j=0, k\4, ((i!=0) + (j!=0)) * (max(0, n+1 - max(4*i,j)) * max(0, k+1 - (4*j+i)) + max(0, n+1 - (4*i+j)) * max(0, k+1 - max(4*j, i)) ))) \\ _Andrew Howroyd_, Mar 11 2024
%Y Diagonal is A189884.
%Y Cf. A189973, A190100, A190103, A190113.
%K nonn,tabl
%O 1,7
%A _R. H. Hardin_, Apr 29 2011
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