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
A320544
(1/8) * number of ways to select 3 distinct points forming a triangle of unsigned area = 1 from a square of grid points with side length n.
Original entry on oeis.org
0, 4, 18, 53, 119, 234, 413, 681, 1047, 1562, 2243, 3101, 4186, 5576, 7231, 9243, 11652, 14518, 17886, 21779, 26191, 31368, 37285, 43919, 51364, 59894, 69338, 79831, 91495, 104336, 118513, 134135, 151072, 169878, 190229, 212185, 236040, 262244, 290317, 320487
Offset: 1
a(1) = 0 because no triangle of area 1 can be formed from the corner points of the [0,1]X[0,1] square.
a(2) = 4 because 3 triangles of area 1 can be formed by connecting the end points of any of the 8 segments of length 1 on the periphery of the [0,2]X[0,2] square to any of the 3 vertices on the opposite side of the grid square, making 8*3 = 24 triangles. Additionally, 4 triangles of the type (0,0),(0,2),(1,2) and another 4 triangles of the type (2,1),(0,1),(1,0) can be selected. 24 + 4 + 4 = 32, a(2) = 32/8 = 4.
A320539
(1/2) * number of ways to select 3 distinct collinear points 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, 1, 4, 4, 10, 22, 10, 21, 42, 76, 20, 39, 70, 120, 186, 35, 65, 112, 184, 279, 412, 56, 100, 166, 264, 390, 566, 772, 84, 146, 236, 367, 532, 759, 1026, 1356, 120, 205, 324, 494, 704, 991, 1326, 1740, 2224, 165, 278, 432, 647, 913, 1271, 1686, 2196, 2793, 3496
Offset: 1
The triangle begins:
0
1 4
4 10 22
10 21 42 76
20 39 70 120 186
35 65 112 184 279 412
56 100 166 264 390 566 772
.
a(2) = T(1,2) = 1, because the grid points on the two longer sides of the rectangle are collinear: (0,0) (0,1) (0,2) and (1,0) (1,1) (2,2).
a(3) = T(2,2) = 4, because there are 8 triples of collinear points:
(0,0) (0,1) (0,2),
(0,0) (1,0) (2,0),
(0,0) (1,1) (2,2),
(0,1) (1,1) (2,1),
(0,2) (1,1) (2,0),
(0,2) (1,2) (2,2),
(1,0) (1,1) (1,2),
(2,0) (2,1) (2,2).
Showing 1-3 of 3 results.
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