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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). 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
T(n,k) = (1/4) * number of ways to select 3 distinct points forming a triangle of unsigned area = 1/2 from a rectangle of grid points with side lengths n and k.
Permutations of the 3 points are not counted separately.
LINKS
EXAMPLE
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.
MAPLE
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
CROSSREFS
Cf. A000217, A115004 (main diagonal), A320539, A320543, A333292.
This triangle is equivalent to the table in A114999.
Sequence in context: A098737 A164654 A225267 * A366612 A072396 A001175
KEYWORD
nonn,tabl
AUTHOR
Hugo Pfoertner, Oct 15 2018
EXTENSIONS
Replaced definition (now a comment) by explicit formula. - N. J. A. Sloane, Feb 04 2020
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)