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A189885
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T(n,k) = number of 1:4:sqrt(17) proportioned triangles on a (n+1) X (k+1) grid.
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5
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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, 20, 32, 36, 56, 56, 36, 32, 20, 24, 44, 48, 80, 96, 80, 48, 44, 24, 28, 56, 68, 104, 136, 136, 104, 68, 56, 28, 32, 68, 88, 140, 176, 192, 176, 140, 88, 68, 32, 36, 80, 108, 180, 232
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OFFSET
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1,7
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LINKS
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FORMULA
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Empirical for columns:
k=1: a(n) = 4*n - 12 for n>2
k=2: a(n) = 12*n - 52 for n>6
k=3: a(n) = 24*n - 136 for n>10
k=4: a(n) = 60*n - 440 for n>15
k=5: a(n) = 108*n - 912 for n>19
k=6: a(n) = 168*n - 1600 for n>23
k=7: a(n) = 240*n - 2552 for n>27
k=8: a(n) = 360*n - 4428 for n>32
k=9: a(n) = 500*n - 6824 for n>36
k=10: a(n) = 660*n - 9820 for n>40
k=11: a(n) = 840*n - 13496 for n>44
k=12: a(n) = 1092*n - 19284 for n>49
k=13: a(n) = 1372*n - 26140 for n>53
k=14: a(n) = 1680*n - 34176 for n>57
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EXAMPLE
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Table starts
..0..0...0...4...8..12..16..20...24...28...32...36...40...44...48...52...56
..0..0...0...8..16..24..32..44...56...68...80...92..104..116..128..140..152
..0..0...0..12..24..36..48..68...88..108..128..152..176..200..224..248..272
..4..8..12..32..56..80.104.140..180..220..260..308..360..412..464..520..580
..8.16..24..56..96.136.176.232..296..360..424..500..584..668..752..844..944
.12.24..36..80.136.192.248.328..420..512..604..712..832..952.1072.1204.1348
.16.32..48.104.176.248.320.424..544..664..784..928.1088.1248.1408.1584.1776
.20.44..68.140.232.328.424.560..716..876.1036.1228.1440.1656.1872.2112.2372
.24.56..88.180.296.420.544.716..912.1116.1320.1564.1832.2108.2384.2692.3024
.28.68.108.220.360.512.664.876.1116.1368.1620.1920.2248.2588.2928.3308.3716
Some solutions for n=6 k=4
..2..0....0..2....6..3....0..4....2..3....1..4....5..3....0..4....1..3....0..0
..2..4....0..1....2..3....0..3....2..4....0..4....1..3....0..0....1..2....0..4
..3..0....4..2....6..2....4..4....6..3....1..0....5..4....1..4....5..3....1..0
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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