login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A259578 Reciprocity array of 2; rectangular, read by antidiagonals. 4
2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 4, 3, 4, 2, 2, 4, 5, 5, 4, 2, 2, 5, 6, 6, 6, 5, 2, 2, 5, 6, 8, 8, 6, 5, 2, 2, 6, 8, 10, 10, 10, 8, 6, 2, 2, 6, 9, 11, 12, 12, 11, 9, 6, 2, 2, 7, 9, 12, 14, 15, 14, 12, 9, 7, 2, 2, 7, 11, 14, 16, 17, 17, 16, 14, 11, 7, 2, 2, 8 (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 = 2 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.

Northwest corner:

2   2   2   2   2   2   2   2   2   2

2   3   3   4   4   5   5   6   6   7

2   3   3   5   6   6   8   9   9   11

2   4   5   6   8   10  11  12  14  16

2   4   6   8   10  12  14  16  18  20

2   5   6   10  12  15  17  20  21  25

MATHEMATICA

x = 2; s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];

TableForm[ Table[s[m, n], {m, 1, 15}, {n, 1, 15}]]

u = Table[s[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten

CROSSREFS

Cf. A259572, A259579, A259580.

Sequence in context: A061357 A270966 A138139 * A266547 A127992 A327390

Adjacent sequences:  A259575 A259576 A259577 * A259579 A259580 A259581

KEYWORD

nonn,easy,tabl

AUTHOR

Clark Kimberling, Jul 17 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 29 16:43 EDT 2020. Contains 334704 sequences. (Running on oeis4.)