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A259575
Reciprocity array of 1; rectangular, read by antidiagonals.
5
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 4, 4, 3, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 6, 7, 7, 6, 4, 1, 1, 4, 7, 8, 10, 8, 7, 4, 1, 1, 5, 8, 10, 11, 11, 10, 8, 5, 1, 1, 5, 9, 12, 13, 15, 13, 12, 9, 5, 1, 1, 6, 10, 13, 15, 16, 16, 15, 13, 10, 6, 1, 1, 6
OFFSET
1,8
COMMENTS
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).
REFERENCES
R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.
LINKS
FORMULA
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.
EXAMPLE
Northwest corner:
1 1 1 1 1 1 1 1 1 1
1 1 2 2 3 3 4 4 5 5
1 2 3 4 5 6 7 8 9 10
1 2 4 6 7 8 10 12 14 15
1 3 5 7 10 11 13 15 17 20
1 3 6 8 11 15 16 18 21 23
1 4 7 10 13 16 21 22 25 28
MATHEMATICA
x = 1; 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 *)
Table[s[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten (* sequence *)
CROSSREFS
KEYWORD
nonn,easy,tabl
AUTHOR
Clark Kimberling, Jul 01 2015
STATUS
approved