A190100 T(n,k) = number of 1:2:sqrt(5) proportioned triangles on a (n+1) X (k+1) grid.

0, 4, 4, 8, 16, 8, 12, 32, 32, 12, 16, 52, 64, 52, 16, 20, 76, 104, 104, 76, 20, 24, 100, 152, 176, 152, 100, 24, 28, 124, 204, 260, 260, 204, 124, 28, 32, 148, 260, 356, 384, 356, 260, 148, 32, 36, 172, 316, 460, 532, 532, 460, 316, 172, 36, 40, 196, 372, 572, 692, 744, 692

Offset

1,2

Links

R. H. Hardin, Table of n, a(n) for n = 1..10025

Formula

Empirical for column k:

k=1: a(n) = 4*n - 4

k=2: a(n) = 24*n - 44 for n>3

k=3: a(n) = 56*n - 132 for n>5

k=4: a(n) = 120*n - 392 for n>8

k=5: a(n) = 204*n - 796 for n>10

k=6: a(n) = 336*n - 1608 for n>13

k=7: a(n) = 496*n - 2716 for n>15

k=8: a(n) = 720*n - 4568 for n>18

k=9: a(n) = 980*n - 6920 for n>20

k=10: a(n) = 1320*n - 10452 for n>23

k=11: a(n) = 1704*n - 14740 for n>25

k=12: a(n) = 2184*n - 20748 for n>28

k=13: a(n) = 2716*n - 27820 for n>30

k=14: a(n) = 3360*n - 37252 for n>33

Example

Table starts
..0...4...8..12...16...20...24...28...32...36...40...44...48...52...56....60
..4..16..32..52...76..100..124..148..172..196..220..244..268..292..316...340
..8..32..64.104..152..204..260..316..372..428..484..540..596..652..708...764
.12..52.104.176..260..356..460..572..688..808..928.1048.1168.1288.1408..1528
.16..76.152.260..384..532..692..868.1052.1248.1448.1652.1856.2060.2264..2468
.20.100.204.356..532..744..976.1236.1512.1808.2116.2436.2764.3096.3432..3768
.24.124.260.460..692..976.1288.1640.2016.2424.2852.3300.3764.4240.4728..5220
.28.148.316.572..868.1236.1640.2104.2600.3144.3716.4324.4956.5612.6288..6980
.32.172.372.688.1052.1512.2016.2600.3224.3916.4644.5424.6236.7088.7968..8876
.36.196.428.808.1248.1808.2424.3144.3916.4776.5684.6664.7688.8768.9888.11052
Some solutions for n=7 k=5
..3..5....0..1....6..3....2..0....5..4....3..3....2..0....3..4....4..1....7..3
..1..3....0..5....4..5....2..1....7..0....3..1....0..2....5..0....0..1....5..3
..7..1....2..1....7..4....4..0....7..5....7..3....3..1....5..5....4..3....7..2

Prog

(PARI) T(n, k)=2*sum(i=0, n\2, sum(j=0, k\2, ((i!=0) + (j!=0))* (max(0, n+1 - max(2*i, j)) * max(0, k+1 - (2*j+i)) + max(0, n+1 - (2*i+j)) * max(0, k+1 - max(2*j, i)) ))) \\ Andrew Howroyd, Mar 11 2024

Crossrefs

Diagonal is A190099.

Cf. A189885.

Sequence in context: A022087 A333149 A095294 * A244421 A030168 A261212

Adjacent sequences: A190097 A190098 A190099 * A190101 A190102 A190103

Keyword

nonn,tabl

Author

R. H. Hardin, May 04 2011

Status

approved