A320541
Triangle read by rows: T(n,k) (1<=k<=n) = Sum_{i=1..n, j=1..k, gcd(i,j)=1} (n+1-i)*(k+1-j).
Original entry on oeis.org
1, 3, 8, 6, 16, 31, 10, 26, 50, 80, 15, 39, 75, 120, 179, 21, 54, 103, 164, 244, 332, 28, 72, 137, 218, 324, 441, 585, 36, 92, 175, 278, 413, 562, 745, 948, 45, 115, 218, 346, 514, 699, 926, 1178, 1463, 55, 140, 265, 420, 623, 846, 1120, 1424, 1768, 2136
Offset: 1
The triangle begins:
1
3 8
6 16 31
10 26 50 80
15 39 75 120 179
21 54 103 164 244 332
28 72 137 218 324 441 585
...
a(1) = 1 because 4 triangles of area 1/2 in a [0 1]X[0 1] square can be formed by cutting the unit square into 2 triangles along the diagonals.
This triangle is equivalent to the table in
A114999.
-
T := proc(m,n) local a,i,j; a:=0;
for i from 1 to m do for j from 1 to n do
if gcd(i,j)=1 then a:=a+(m+1-i)*(n+1-j); fi; od: od: a; end;
for m from 1 to 12 do lprint([seq(T(m,n),n=1..m)]); od: # N. J. A. Sloane, Feb 04 2020
Replaced definition (now a comment) by explicit formula. -
N. J. A. Sloane, Feb 04 2020
A320540
(1/4) * number of ways to select 3 distinct collinear points from a square of grid points with side length n.
Original entry on oeis.org
0, 2, 11, 38, 93, 206, 386, 678, 1112, 1748, 2583, 3768, 5253, 7172, 9630, 12720, 16370, 20910, 26169, 32566, 40139, 48962, 58900, 70710, 84096, 99284, 116469, 136116, 157671, 182436, 209436, 239596, 272976, 309630, 350035, 395346, 444021, 496890, 554402, 617906
Offset: 1
a(2) = 2 because there are 8 triples of collinear points in the square [0 2] X [0 2]: The 2*3 lines of x=0,1,2 and y=0,1,2 and the 2 diagonals.
(1/2)* diagonal of triangle
A320539.
A320543
(1/2) * number of ways to select 3 distinct points forming a triangle of unsigned area = 1 from a rectangle of grid points with side lengths j and k, written as triangle T(j,k), j<=k.
Original entry on oeis.org
0, 3, 16, 8, 35, 72, 15, 62, 125, 212, 24, 95, 190, 319, 476, 35, 136, 269, 450, 669, 936, 48, 183, 360, 601, 892, 1245, 1652, 63, 238, 467, 776, 1149, 1602, 2123, 2724, 80, 299, 584, 967, 1430, 1991, 2636, 3379, 4188, 99, 368, 717, 1186, 1751, 2436, 3223, 4130, 5117, 6248
Offset: 1
The triangle begins:
0
3 16
8 35 72
15 62 125 212
24 95 190 319 476
35 136 269 450 669 936
.
a(2) = T(1,2) = 3 = 6/2 because the following 6 triangles of area 1 can be made by selecting 3 grid points from the [0,1]X[0,2] rectangle:
(0,0) (0,2) (1,0),
(0,0) (0,2) (1,1),
(0,0) (0,2) (1,2),
(0,0) (1,0) (1,2),
(0,1) (1,0) (1,2),
(0,2) (1,0) (1,2).
Showing 1-3 of 3 results.
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