A190113 - OEIS

%I #11 Mar 11 2024 18:06:14

%S 0,0,0,0,0,0,0,4,4,0,0,8,16,8,0,0,12,28,28,12,0,0,16,44,48,44,16,0,0,

%T 20,60,76,76,60,20,0,0,24,76,108,120,108,76,24,0,0,28,92,140,172,172,

%U 140,92,28,0,0,32,108,176,224,248,224,176,108,32,0,0,36,124,212,284,328,328,284

%N T(n,k) = number of 2:3:sqrt(13) proportioned triangles on a (n+1) X (k+1) grid.

%H R. H. Hardin, <a href="/A190113/b190113.txt">Table of n, a(n) for n = 1..9797</a>

%F Empirical for columns:

%F k=2: a(n) = 4*n - 8 for n>1

%F k=3: a(n) = 16*n - 36 for n>3

%F k=4: a(n) = 36*n - 112 for n>6

%F k=5: a(n) = 60*n - 196 for n>6

%F k=6: a(n) = 108*n - 472 for n>11

%F k=7: a(n) = 160*n - 768 for n>11

%F k=8: a(n) = 228*n - 1232 for n>14

%F k=9: a(n) = 320*n - 1948 for n>16

%F k=10: a(n) = 432*n - 2956 for n>19

%F k=11: a(n) = 552*n - 4028 for n>19

%F k=12: a(n) = 720*n - 5900 for n>24

%F k=13: a(n) = 896*n - 7848 for n>24

%F k=14: a(n) = 1100*n - 10348 for n>27

%e Table starts

%e .0..0...0...0...0...0...0....0....0....0....0....0....0....0....0....0....0

%e .0..0...4...8..12..16..20...24...28...32...36...40...44...48...52...56...60

%e .0..4..16..28..44..60..76...92..108..124..140..156..172..188..204..220..236

%e .0..8..28..48..76.108.140..176..212..248..284..320..356..392..428..464..500

%e .0.12..44..76.120.172.224..284..344..404..464..524..584..644..704..764..824

%e .0.16..60.108.172.248.328..420..516..616..720..824..932.1040.1148.1256.1364

%e .0.20..76.140.224.328.440..568..704..848.1000.1152.1312.1472.1632.1792.1952

%e .0.24..92.176.284.420.568..736..916.1108.1312.1520.1740.1964.2188.2416.2644

%e .0.28.108.212.344.516.704..916.1152.1404.1672.1952.2248.2552.2860.3176.3492

%e .0.32.124.248.404.616.848.1108.1404.1720.2056.2412.2788.3176.3576.3988.4404

%e Some solutions for n=7 k=5

%e ..3..5....1..2....4..3....1..4....0..3....4..4....4..0....3..3....4..4....3..0

%e ..3..3....1..4....1..0....1..0....2..5....4..1....2..0....0..3....1..4....3..2

%e ..6..5....4..2....2..5....7..4....3..0....6..4....4..3....3..5....4..2....6..0

%o (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,2*j)) * max(0, k+1 - (3*j+2*i)) + max(0, n+1 - (3*i+2*j)) * max(0, k+1 - max(3*j, 2*i)) ))) \\ _Andrew Howroyd_, Mar 11 2024

%Y Diagonal is A190112.

%Y Cf. A189885.

%K nonn,tabl

%O 1,8

%A _R. H. Hardin_, May 04 2011