|
%I #13 Mar 11 2024 13:34:15
%S 0,4,4,8,16,8,12,32,32,12,16,52,64,52,16,20,76,104,104,76,20,24,100,
%T 152,176,152,100,24,28,124,204,260,260,204,124,28,32,148,260,356,384,
%U 356,260,148,32,36,172,316,460,532,532,460,316,172,36,40,196,372,572,692,744,692
%N T(n,k) = number of 1:2:sqrt(5) proportioned triangles on a (n+1) X (k+1) grid.
%H R. H. Hardin, <a href="/A190100/b190100.txt">Table of n, a(n) for n = 1..10025</a>
%F Empirical for column k:
%F k=1: a(n) = 4*n - 4
%F k=2: a(n) = 24*n - 44 for n>3
%F k=3: a(n) = 56*n - 132 for n>5
%F k=4: a(n) = 120*n - 392 for n>8
%F k=5: a(n) = 204*n - 796 for n>10
%F k=6: a(n) = 336*n - 1608 for n>13
%F k=7: a(n) = 496*n - 2716 for n>15
%F k=8: a(n) = 720*n - 4568 for n>18
%F k=9: a(n) = 980*n - 6920 for n>20
%F k=10: a(n) = 1320*n - 10452 for n>23
%F k=11: a(n) = 1704*n - 14740 for n>25
%F k=12: a(n) = 2184*n - 20748 for n>28
%F k=13: a(n) = 2716*n - 27820 for n>30
%F k=14: a(n) = 3360*n - 37252 for n>33
%e Table starts
%e ..0...4...8..12...16...20...24...28...32...36...40...44...48...52...56....60
%e ..4..16..32..52...76..100..124..148..172..196..220..244..268..292..316...340
%e ..8..32..64.104..152..204..260..316..372..428..484..540..596..652..708...764
%e .12..52.104.176..260..356..460..572..688..808..928.1048.1168.1288.1408..1528
%e .16..76.152.260..384..532..692..868.1052.1248.1448.1652.1856.2060.2264..2468
%e .20.100.204.356..532..744..976.1236.1512.1808.2116.2436.2764.3096.3432..3768
%e .24.124.260.460..692..976.1288.1640.2016.2424.2852.3300.3764.4240.4728..5220
%e .28.148.316.572..868.1236.1640.2104.2600.3144.3716.4324.4956.5612.6288..6980
%e .32.172.372.688.1052.1512.2016.2600.3224.3916.4644.5424.6236.7088.7968..8876
%e .36.196.428.808.1248.1808.2424.3144.3916.4776.5684.6664.7688.8768.9888.11052
%e Some solutions for n=7 k=5
%e ..3..5....0..1....6..3....2..0....5..4....3..3....2..0....3..4....4..1....7..3
%e ..1..3....0..5....4..5....2..1....7..0....3..1....0..2....5..0....0..1....5..3
%e ..7..1....2..1....7..4....4..0....7..5....7..3....3..1....5..5....4..3....7..2
%o (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
%Y Diagonal is A190099.
%Y Cf. A189885.
%K nonn,tabl
%O 1,2
%A _R. H. Hardin_, May 04 2011
| 