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A320541
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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).
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7
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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
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CROSSREFS
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This triangle is equivalent to the table in A114999.
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KEYWORD
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AUTHOR
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EXTENSIONS
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Replaced definition (now a comment) by explicit formula. - N. J. A. Sloane, Feb 04 2020
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STATUS
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approved
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