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A259572 Reciprocity array of 0; rectangular, read by antidiagonals. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,12

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). 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

0             A259572      A259573      A259574

1             A259575      A259576      A259577

2             A259578      A259579      A249580

3             A259581      A259582      A249583

REFERENCES

R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989, pages 90-94.

LINKS

Clark Kimberling, Antidiagonals n=1..60, flattened

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.

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.

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A259573, A259574.

Sequence in context: A009108 A016537 A106385 * A027413 A019509 A071484

Adjacent sequences:  A259569 A259570 A259571 * A259573 A259574 A259575

KEYWORD

nonn,easy,tabl

AUTHOR

Clark Kimberling, Jun 30 2015

STATUS

approved

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Last modified May 29 21:22 EDT 2017. Contains 287257 sequences.