A259572 -id:A259572 - cp's OEIS frontend

A259574 Sum of numbers in the n-th antidiagonal of the reciprocity array of 0.

0, 1, 4, 11, 22, 42, 66, 104, 150, 211, 280, 377, 474, 604, 750, 916, 1096, 1323, 1548, 1831, 2122, 2446, 2794, 3212, 3620, 4087, 4590, 5141, 5698, 6360, 6990, 7728, 8484, 9289, 10156, 11091, 12006, 13042, 14122, 15280, 16420, 17727, 18984, 20401, 21852

Offset: 1

Clark Kimberling, Jun 30 2015

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/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.

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

Clark Kimberling, Table of n, a(n) for n = 1..500

Cf. A259572, A259573.

seq(add(add(floor(n*k/m),k=0..m-1),m=1..n), n=1..100); # Robert Israel, Jul 06 2015

f[n_] := Sum[Floor[n*k/m], {m, n}, {k, 0, m - 1}]; Array[f, 50]

a(n) = {r=0;for(m=1,n,for(k=0,m-1,r=r+floor((n*k)/m)));return(r);} main(size)={return(vector(size,n,a(n)));} /* Anders Hellström, Jul 07 2015 */

a(n) = Sum_{m=1..n} Sum_{k=0..m-1} floor(n*k/m).

A259575 Reciprocity array of 1; rectangular, read by antidiagonals.

Original entry on oeis.org

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

Clark Kimberling, Jul 01 2015

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/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.

			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  13  14
  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

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

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

Cf. A259572, A259576, A259577.

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 *)

T(m,n) = Sum_{k=0..m-1} [(n*k+x)/m] = Sum_{k=0..n-1} [(m*k+x)/n], where x = 1 and [ ] = floor.

A259578 Reciprocity array of 2; rectangular, read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 17 2015

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/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.

			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

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

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

Cf. A259572, A259579, A259580.

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

T(m,n) = Sum_{k=0..m-1} [(n*k+x)/m] = Sum_{k=0..n-1} [(m*k+x)/n], 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.

A259580 Sum of numbers in the n-th antidiagonal of the reciprocity array of 2.

Original entry on oeis.org

2, 5, 8, 17, 30, 50, 78, 116, 162, 227, 300, 389, 498, 628, 766, 940, 1128, 1347, 1584, 1855, 2146, 2486, 2838, 3236, 3660, 4135, 4626, 5177, 5754, 6392, 7050, 7776, 8524, 9353, 10204, 11127, 12078, 13114, 14170, 15328, 16500, 17775, 19068, 20461, 21900

Offset: 1

Clark Kimberling, Jul 17 2015

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/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.

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

Clark Kimberling, Table of n, a(n) for n = 1..500

Cf. A259572, A259578, A259579.

x = 2;  v[n_] := Sum[Sum[Floor[(n*k + x)/m], {k, 0, m - 1}], {m, 1, n}];
Table[v[n], {n, 1, 120}]

a(n) = Sum_{m=1..n} Sum_{k=0..m-1} floor((n*k + x)/m), where x = 2.

A259581 Reciprocity array of 3; rectangular, read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 15 2015

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/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.

			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   4   6   6   9   10  12  12  15  16
  3   5   7   9   10  13  15  17  19  20
  3   5   9   10  13  15  18  20  24  25

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

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

Cf. A259572, A259582, A259583.

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 *)

T(m,n) = Sum_{k=0..m-1} [(n*k+x)/m] = Sum_{k=0..n-1} [(m*k+x)/n], 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.

A259583 Sum of numbers in the n-th antidiagonal of the reciprocity array of 3.

Original entry on oeis.org

3, 6, 13, 19, 34, 55, 84, 120, 174, 231, 310, 399, 510, 634, 786, 948, 1144, 1359, 1602, 1863, 2176, 2496, 2860, 3256, 3680, 4147, 4662, 5189, 5782, 6412, 7080, 7792, 8574, 9369, 10228, 11151, 12114, 13132, 14230, 15344, 16540, 17805, 19110, 20481, 21948

Offset: 1

Clark Kimberling, Jul 15 2015

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/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.

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

Clark Kimberling, Table of n, a(n) for n = 1..499

Cf. A259572, A259581, A259582.

x = 3;  v[n_] := Sum[Sum[Floor[(n*k + x)/m], {k, 0, m - 1}], {m, 1, n}];
Table[v[n], {n, 1, 120}]

a(n) = Sum_{m=1..n} Sum_{k=0..m-1} floor((n*k + x)/m), where x = 3.

A259573 Number of distinct differences in row n of the reciprocity array of 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 3, 6, 5, 6, 3, 8, 3, 6, 7, 8, 3, 8, 3, 8, 9, 6, 3, 12, 5, 6, 7, 10, 3, 14, 3, 10, 9, 6, 9, 14, 3, 6, 9, 12, 3, 12, 3, 12, 11, 6, 3, 18, 5, 10, 9, 12, 3, 12, 9, 14, 9, 6, 3, 22, 3, 6, 13, 12, 9, 14, 3, 12, 9, 14, 3, 18, 3, 6, 13, 12, 9, 16

Offset: 1

Clark Kimberling, Jun 30 2015

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/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).

			In the array at A259572, row 4 is (0,2,3,6,6,8,9,12,12,14,15,...), with differences (2,1,3,0,2,1,3,0,2,1,3,0, ...), and distinct differences {0,1,2,3}, so that a(4) = 4. Example corrected by _Antti Karttunen_, Nov 30 2021

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

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A249572, A259574.

x = 0;  s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
t[m_] := Table[s[m, n], {n, 1, 1000}];
u = Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]  (* A259573 *)

A259572(m,n) = ((m*n - m - n + gcd(m,n))/2); \\ After Witold Dlugosz's formula for A259572.
A259573(n) = #Set(vector(n,k,A259572(n,1+k)-A259572(n,k))); \\ Antti Karttunen, Nov 30 2021

A259576 Number of distinct differences in row n of the reciprocity array of 1.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 3, 4, 3, 6, 3, 6, 3, 6, 5, 6, 3, 8, 3, 8, 5, 6, 3, 10, 5, 6, 5, 10, 3, 10, 3, 8, 5, 6, 7, 14, 3, 6, 5, 12, 3, 12, 3, 10, 11, 6, 3, 14, 5, 10, 5, 10, 3, 12, 9, 12, 5, 6, 3, 18, 3, 6, 11, 10, 9, 14, 3, 10, 5, 16, 3, 18, 3, 6, 9, 10, 7, 14, 3

Offset: 1

Clark Kimberling, Jul 01 2015

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/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.

			In the array at A259575, row 6 is (1,3,6,8,11,15,16,18,...), with differences (2,3,2,3,4,1,2,...), and distinct differences {1,2,3,4}, so that a(6) = 4.

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

Antti Karttunen, Table of n, a(n) for n = 1..5000

Cf. A249572, A249573, A259575.

x = 1;  s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
t[m_] := Table[s[m, n], {n, 1, 1000}];
u = Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]  (* A259576 *)

A259575sq(m,n) = sum(k=0,m-1,(1+(n*k))\m);
A259576(n) = #Set(vector(n,k,A259575sq(n,1+k)-A259575sq(n,k))); \\ Antti Karttunen, Mar 02 2023

A259577 Sum of numbers in the n-th antidiagonal of the reciprocity array of 1.

Original entry on oeis.org

1, 2, 6, 13, 26, 44, 72, 108, 156, 215, 290, 381, 486, 610, 758, 924, 1112, 1329, 1566, 1839, 2134, 2456, 2816, 3220, 3640, 4099, 4608, 5153, 5726, 6368, 7020, 7744, 8504, 9305, 10180, 11103, 12042, 13060, 14146, 15296, 16460, 17739, 19026, 20421, 21876

Offset: 1

Clark Kimberling, Jul 01 2015

nonn

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/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.

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

Clark Kimberling, Table of n, a(n) for n = 1..500

Cf. A259572, A259574, A259575.

f[n_] := Sum[Floor[(n*k + 1)/m], {m, n}, {k, 0, m - 1}]; Array[f, 50]

a(n)=x=1;r=0;for(m=1,n,for(k=0,m-1,r=r+floor((n*k+x)/m)));return(r);
main(size)=return(vector(size,n,a(n))) \\ Anders Hellström, Jul 06 2015

a(n)=sum(m=1,n, sum(k=0,m-1, (n*k+1)\m)) \\ Charles R Greathouse IV, Mar 22 2017

a(n) = Sum_{m=1..n} Sum_{k=0..m-1} floor((n*k + x)/m), where x = 1.

a(n) = n^3 / 4 + O(n^2). - Charles R Greathouse IV, Mar 22 2017

A259579 Number of distinct differences in row n of the reciprocity array of 2.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 4, 5, 4, 3, 6, 3, 4, 5, 6, 3, 6, 3, 6, 7, 6, 3, 10, 3, 6, 7, 8, 3, 12, 3, 8, 9, 6, 5, 12, 3, 6, 9, 10, 3, 12, 3, 10, 9, 6, 3, 16, 5, 8, 9, 10, 3, 10, 5, 10, 9, 6, 3, 20, 3, 6, 9, 10, 5, 14, 3, 10, 9, 12, 3, 16, 3, 6, 11, 10, 9, 14, 3, 14

Offset: 1

Clark Kimberling, Jul 17 2015

The "reciprocity law" that Sum_{k=0..m} [(n*k+x)/m] = Sum_{k=0..n} [(m*k+x)/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.

			In the array at A259578, row 6 is (2,5,6,10,12,15,17,20,21,25,27,...), with differences (3,1,4,2,3,2,3,1,4,2,...), and distinct differences {1,2,3,4}, so that a(6) = 4.

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

Cf. A249572, A249577, A259580.

x = 2;  s[m_, n_] := Sum[Floor[(n*k + x)/m], {k, 0, m - 1}];
t[m_] := Table[s[m, n], {n, 1, 1000}];
u = Table[Length[Union[Differences[t[m]]]], {m, 1, 120}]

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A259574 Sum of numbers in the n-th antidiagonal of the reciprocity array of 0.

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A259575 Reciprocity array of 1; rectangular, read by antidiagonals.

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A259578 Reciprocity array of 2; rectangular, read by antidiagonals.

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A259580 Sum of numbers in the n-th antidiagonal of the reciprocity array of 2.

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A259581 Reciprocity array of 3; rectangular, read by antidiagonals.

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A259583 Sum of numbers in the n-th antidiagonal of the reciprocity array of 3.

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A259573 Number of distinct differences in row n of the reciprocity array of 0.

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