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A259572
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Reciprocity array of 0; rectangular, read by antidiagonals.
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12
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0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 3, 2, 0, 0, 2, 3, 3, 2, 0, 0, 3, 4, 6, 4, 3, 0, 0, 3, 6, 6, 6, 6, 3, 0, 0, 4, 6, 8, 10, 8, 6, 4, 0, 0, 4, 7, 9, 10, 10, 9, 7, 4, 0, 0, 5, 9, 12, 12, 15, 12, 12, 9, 5, 0, 0, 5, 9, 12, 14, 15, 15, 14, 12, 9, 5, 0, 0, 6, 10
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OFFSET
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1,12
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COMMENTS
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The "reciprocity law" that Sum{[(n*k+x)/m]:k = 0..m} = Sum{[(m*k+x)/n]:k = 0..n} where x is a real number and m and n are positive integers, is proved in Section 3.5 of Concrete Mathematics (see References). For every x, the reciprocity array is symmetric, and the principal diagonal consists primarily of triangular numbers, A000217.
In the following guide, the sequence in column 3 is the number of distinct terms in the difference sequence of row n of the reciprocity array of x; sequence in column 4 is the sum of numbers in the n-th antidiagonal of the array.
x array differences sums
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REFERENCES
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R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
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LINKS
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FORMULA
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T(m,n) = Sum{[(n*k+x)/m] : k = 0..m-1} = Sum{[(m*k+x)/n] : k = 0..n-1}, where x = 1 and [ ] = floor.
Note that if [x] = [y], then [(n*k+x)/m] = [(n*k+y/m], so that the reciprocity arrays for x and y are identical in this case.
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EXAMPLE
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Northwest corner:
0 0 0 0 0 0 0 0 0 0
0 1 1 2 2 3 3 4 4 5
0 1 3 3 4 6 6 7 9 9
0 2 3 6 6 8 9 12 12 14
0 2 4 6 10 10 12 14 16 20
0 3 6 8 10 15 15 18 21 23
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MATHEMATICA
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x = 0; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
TableForm[ Table[s[m, n], {m, 1, 15}, {n, 1, 15}]] (* array *)
u = Table[s[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten (* sequence *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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