A190103 T(n,k) = number of 1:3:sqrt(10) proportioned triangles on a (n+1) X (k+1) grid.

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, 36, 20, 24, 48, 92, 116, 116, 92, 48, 24, 28, 60, 124, 164, 168, 164, 124, 60, 28, 32, 72, 156, 220, 240, 240, 220, 156, 72, 32, 36, 84, 192, 276, 324, 344, 324, 276, 192, 84, 36, 40, 96

Offset

1,4

Links

R. H. Hardin, Table of n, a(n) for n = 1..9786

Formula

Empirical for columns:

k=1: a(n) = 4*n - 8 for n>1

k=2: a(n) = 12*n - 36 for n>4

k=3: a(n) = 40*n - 168 for n>8

k=4: a(n) = 80*n - 400 for n>11

k=5: a(n) = 132*n - 768 for n>14

k=6: a(n) = 224*n - 1588 for n>18

k=7: a(n) = 336*n - 2712 for n>21

k=8: a(n) = 468*n - 4200 for n>24

k=9: a(n) = 660*n - 6732 for n>28

k=10: a(n) = 880*n - 9884 for n>31

k=11: a(n) = 1128*n - 13740 for n>34

k=12: a(n) = 1456*n - 19476 for n>38

k=13: a(n) = 1820*n - 26264 for n>41

k=14: a(n) = 2220*n - 34208 for n>44

Example

Table starts
..0..0...4...8..12..16...20...24...28...32...36...40...44...48...52...56...60
..0..0...8..16..24..36...48...60...72...84...96..108..120..132..144..156..168
..4..8..24..44..64..92..124..156..192..232..272..312..352..392..432..472..512
..8.16..44..80.116.164..220..276..340..412..484..560..640..720..800..880..960
.12.24..64.116.168.240..324..408..504..612..720..836..960.1084.1212.1344.1476
.16.36..92.164.240.344..464..588..732..892.1056.1232.1420.1612.1812.2020.2232
.20.48.124.220.324.464..624..792..988.1204.1428.1672.1932.2200.2480.2772.3072
.24.60.156.276.408.588..792.1008.1260.1536.1824.2140.2476.2824.3192.3576.3972
.28.72.192.340.504.732..988.1260.1584.1936.2304.2712.3144.3592.4072.4572.5088
.32.84.232.412.612.892.1204.1536.1936.2368.2820.3328.3864.4420.5020.5644.6288
Some solutions for n=7 k=5
..0..3....3..3....0..0....2..1....3..5....1..1....2..4....7..0....4..2....0..1
..0..1....3..2....0..2....2..0....0..2....1..3....2..1....1..0....1..2....0..2
..6..3....6..3....6..0....5..1....4..4....7..1....3..4....7..2....4..1....3..1

Prog

(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

Crossrefs

Diagonal is A190102.

Cf. A189885.

Sequence in context: A298518 A021251 A160207 * A176714 A055951 A165032

Adjacent sequences: A190100 A190101 A190102 * A190104 A190105 A190106

Keyword

nonn,tabl

Author

R. H. Hardin, May 04 2011

Status

approved