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A281726
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Triangular array read by rows: T(n,k) is the number of elements in an n X k matrix that will be assigned the same value whether the integers from 1 to n*k are assigned to elements in row-major order or column-major order.
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2
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1, 2, 2, 3, 2, 3, 4, 2, 2, 4, 5, 2, 3, 2, 5, 6, 2, 2, 2, 2, 6, 7, 2, 3, 4, 3, 2, 7, 8, 2, 2, 2, 2, 2, 2, 8, 9, 2, 3, 2, 5, 2, 3, 2, 9, 10, 2, 2, 4, 2, 2, 4, 2, 2, 10, 11, 2, 3, 2, 3, 6, 3, 2, 3, 2, 11, 12, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 12, 13, 2, 3, 4, 5, 2, 7, 2, 5, 4, 3, 2, 13
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OFFSET
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1,2
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COMMENTS
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T(n,k) is also the number of lattice points that lie on a line segment from (0,0) to (n-k,k-1). Thus, row n of the triangle lists, for each of the n 1st-quadrant lattice points P whose Manhattan distance from the origin is n-1, the number of lattice points on a line segment from the origin to P.
E.g., for n = 5, the 5 1st-quadrant lattice points whose Manhattan distance from the origin is 4 are (0,4), (1,3), (2,2), (3,1), and (4,0), and a line segment drawn from the origin to each of these points will intersect 5, 2, 3, 2, and 5 lattice points, respectively; { 5, 2, 3, 2, 5 } is row 5 of the triangle. (End)
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LINKS
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FORMULA
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EXAMPLE
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For n=3 and k=2, the matrix will be
1 2 and 1 4
3 4 2 5
5 6 3 6
and there are 2 identical terms (1 and 6).
Triangle begins:
1;
2, 2;
3, 2, 3;
4, 2, 2, 4;
5, 2, 3, 2, 5;
6, 2, 2, 2, 2, 6;
...
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MAPLE
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T:= (n, k)-> add(add(`if`(j+k*(i-1)=i+n*(j-1), 1, 0), i=1..n), j=1..k):
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MATHEMATICA
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Array[1+GCD[#, Range[0, #]]&, 20, 0] (* Paolo Xausa, Dec 08 2023 *)
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PROG
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(PARI) a(n, k) = {ml = matrix(n, k, i, j, ((i-1)*k+j)); mc = matrix(n, k, i, j, ((j-1)*n+i)); sum(i=1, n, sum(j=1, k, ml[i, j] == mc[i, j])); }
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CROSSREFS
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Main diagonal and column k=1 give A000027.
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KEYWORD
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AUTHOR
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STATUS
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approved
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