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A259581 Reciprocity array of 3; rectangular, read by antidiagonals. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).  See A259572 for a guide to related sequences.

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 = 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.

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A259572, A259582, A259583.

Sequence in context: A162844 A115787 A300959 * A105592 A210745 A187471

Adjacent sequences:  A259578 A259579 A259580 * A259582 A259583 A259584

KEYWORD

nonn,easy,tabl

AUTHOR

Clark Kimberling, Jul 15 2015

STATUS

approved

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Last modified May 31 13:01 EDT 2020. Contains 334748 sequences. (Running on oeis4.)