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A259581
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Reciprocity array of 3; rectangular, read by antidiagonals.
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4
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3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 6, 4, 3, 3, 5, 6, 6, 5, 3, 3, 5, 7, 6, 7, 5, 3, 3, 6, 9, 9, 9, 9, 6, 3, 3, 6, 9, 10, 10, 10, 9, 6, 3, 3, 7, 10, 12, 13, 13, 12, 10, 7, 3, 3, 7, 12, 12, 15, 15, 15, 12, 12, 7, 3, 3, 8, 12, 15, 17, 18, 18, 17, 15, 12, 8, 3
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OFFSET
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1,1
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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). See A259572 for a guide to related sequences.
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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 = 3 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.
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EXAMPLE
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Northwest corner:
3 3 3 3 3 3 3 3 3 3
3 3 4 4 5 5 6 6 7 7
3 4 6 6 7 9 9 10 12 12
3 5 7 9 10 12 12 15 16 18
3 5 9 10 13 15 17 19 20 23
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MATHEMATICA
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x = 3; 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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