A144112 -id:A144112 - cp's OEIS frontend

A185910 Array: T(n,k) = n^2 + k - 1, by antidiagonals.

1, 2, 4, 3, 5, 9, 4, 6, 10, 16, 5, 7, 11, 17, 25, 6, 8, 12, 18, 26, 36, 7, 9, 13, 19, 27, 37, 49, 8, 10, 14, 20, 28, 38, 50, 64, 9, 11, 15, 21, 29, 39, 51, 65, 81, 10, 12, 16, 22, 30, 40, 52, 66, 82, 100, 11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 121, 12, 14, 18, 24, 32, 42, 54, 68, 84, 102, 122, 144, 13, 15, 19, 25, 33, 43, 55, 69, 85, 103, 123, 145, 169, 14, 16, 20, 26, 34, 44, 56, 70, 86, 104, 124, 146, 170, 196

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185911 < A185910 < A185912 < A185913 < ... (See A144112 for definitions of weight array and accumulation array.)

			Northwest corner:
   1,  2,  3,  4,  5
   4,  5,  6,  7,  8
   9, 10, 11, 12, 13
  16, 17, 18, 19, 20

G. C. Greubel, Table of n, a(n) for the first 50 antidiagonals, flattened

Cf. A144112, A185910, A185912.

(* This program generates the array A185910, its accumulation array A185812, and its weight array A185911. *)
f[n_,0]:=0;f[0,k_]:=0;
f[n_,k_]:=n^2+k-1;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185910 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]]  (* formula for A185812 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A185911 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = n^2 + k - 1, k >= 1, n >= 1.

A185957 Second accumulation array of the array min{n,k}, by antidiagonals.

Original entry on oeis.org

1, 3, 3, 6, 10, 6, 10, 21, 21, 10, 15, 36, 46, 36, 15, 21, 55, 81, 81, 55, 21, 28, 78, 126, 146, 126, 78, 28, 36, 105, 181, 231, 231, 181, 105, 36, 45, 136, 246, 336, 371, 336, 246, 136, 45, 55, 171, 321, 461, 546, 546, 461, 321, 171, 55, 66, 210, 406, 606, 756, 812, 756

Offset: 1

Clark Kimberling, Feb 07 2011

A member of the accumulation chain
... < A003982 < A003783 < A115262 < A185957 <...,
where A003783(n,k)=min{n,k}.  See A144112 for the definition of accumulation array.
A185957 also gives the symmetric matrix based on the triangular numbers s=(1,3,6,10,15,....; viz, let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s.  Let T' be the transpose of T.  Then A185957 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A202678 for characteristic polynomials of principal submatrices of M.

			Northwest corner:
1....3....6....10...15
3....10...21...36...55
6....21...46...81...126
10...36...81...146..231

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A003783, A115262.

U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k (k + 1)/2, {k, 1, 15}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}] (* A000292 *)
Table[m[1, j], {j, 1, 12}] (* A000217 *)
Table[m[2, j], {j, 1, 12}] (* A014105 *)
Table[m[j, j], {j, 1, 12}] (* A024166 *)
Table[m[j, j + 1], {j, 1, 12}] (* A112851 *)
Table[Sum[m[i, n + 1 - i], {i, 1, n}], {n, 1, 12}] (* A001769 *)

A144148 Weight array W={w(i,j)} of the Wythoff array A035513.

Original entry on oeis.org

1, 1, 3, 1, 2, 2, 2, 3, 1, 3, 3, 5, 2, 2, 3, 5, 8, 3, 3, 2, 2, 8, 13, 5, 5, 3, 1, 3, 13, 21, 8, 8, 5, 2, 2, 2, 21, 34, 13, 13, 8, 3, 3, 1, 3, 34, 55, 21, 21, 13, 5, 5, 2, 2, 3, 55, 89, 34, 34, 21, 8, 8, 3, 3, 2, 2, 89, 144, 55, 55, 34, 13, 13, 5, 5, 3, 1, 3, 144, 233, 89, 89, 55, 21, 21, 8, 8, 5, 2, 2, 3

Offset: 1

Clark Kimberling, Sep 11 2008

In general, let w(i,j) be the weight of the unit square labeled by its northeast vertex (i,j) and for each (m,n), define S(m,n) = Sum_{i=1..m} Sum_{j=1..n} w(i,j).
Then S(m,n) is the weight of the rectangle [0,m]x[0,n]. As in A144112, we call W the weight array of S, and S the accumulation array of W, which can be derived from S as follows:
(1) extend S by defining S(i,j)=0 if i=0 or j=0; and
(2) then w(m,n) = s(m,n) + s(m-n,n-1) - s(m,n-1) - s(n,m-1) for m>=1, n>=1.
For the case at hand, S is the Wythoff array, A035513.  These arrays form a chain:
... ->A144148->A035513->A185737-> ...  Every term of this array is a Fibonacci number.

			Corner:
    1  1  1  2  3   5   8  13  21  34   55   89
    3  2  3  5  8  13  21  34  55  89  144  233
    2  1  2  3  5   8  13  21  34  55   89  144
    3  2  3  5  8  13  21  34  55  89  144  233
    3  2  3  5  8  13  21  34  55  89  144  233
    2  1  2  3  5   8  13  21  34  55   89  144
    3  2  3  5  8  13  21  34  55  89  144  233
    2  1  2  3  5   8  13  21  34  55   89  144
    3  2  3  5  8  13  21  34  55  89  144  233
    3  2  3  5  8  13  21  34  55  89  144  233
    2  1  2  3  5   8  13  21  34  55   89  144
    3  2  3  5  8  13  21  34  55  89  144  233

Cf. A000045, A000201, A001950, A035513, A144112, A185737, A185778.

s[n_, k_] := Fibonacci[k + 1]  Floor[n*GoldenRatio] + (n - 1)  Fibonacci[k];
Grid[Table[s[n, k], {n, 1, 12}, {k, 1, 12}]]   (* A035513 *)
s[0, k_] := 0; s[n_, 0] = 0;
w[m_, n_] := s[m, n] + s[m - 1, n - 1] - s[m, n - 1] - s[m - 1, n];
Grid[Table[w[n, k], {n, 1, 12}, {k, 1, 12}]] (* array *)
Table[w[k, m - k], {m, 2, 14}, {k, 1, m - 1}] // Flatten (* sequence *)

s(n, k) = if ((n<=0) || (k<=0), 0, (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k)); \\ A035513
w(n, k) = s(n,k)+s(n-1,k-1)-s(n,k-1)-s(n-1,k); \\ Michel Marcus, Feb 02 2025

For m>3, if the row number is m of form floor(h*r+1), where r=(1+sqrt(5))/2, then
 (row m)=(row 2); otherwise, (row m)=(row 3).
row n: (3,2,3,5,8,13,21,...) if n>1 is in the lower Wythoff sequence, A000201.
row n: (2,1,2,3,5,8,13,21,...) if n is in the upper Wythoff sequence, A001950.

Corrected and extended by Michel Marcus, Feb 02 2025
Some of the content of the duplicate (and now dead) sequence A185736 has been merged into this entry. - N. J. A. Sloane, Feb 15 2025
Edited by Clark Kimberling, Feb 16 2025

A144257 Weight array of A086270.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 2, 1, 0, 5, 3, 2, 1, 0, 6, 4, 3, 2, 1, 0, 7, 5, 4, 3, 2, 1, 0, 8, 6, 5, 4, 3, 2, 1, 0, 9, 7, 6, 5, 4, 3, 2, 1, 0, 10, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 12, 11, 10, 9, 8, 7, 6, 5

Offset: 1

Clark Kimberling, Sep 16 2008

For the definition of weight array, see A144112.
From Gary W. Adamson, Feb 18 2010: (Start)
Identical to an infinite lower triangular matrix with (1,2,3,...) in every column but the leftmost column shifted one row upwards, giving:
  1;
  2, 0;
  3, 1, 0;
  4, 2, 1, 0;
  5, 3, 2, 1, 0;
  ...
Let the triangle = M. Row sums = A000124; M * [1,2,3,...] = A050407 starting with offset 3: (1, 2, 5, 11, 21, 36, ...); and lim_{n->inf} M^n = the odd-indexed Fibonacci numbers, A001519: (1, 2, 5, 13, ...). (End)

			Northwest corner:
  1 2 3 4 5 6 7 8 9
  0 1 2 3 4 5 6 7 8
  0 1 2 3 4 5 6 7 8
  0 1 2 3 4 5 6 7 8
  0 1 2 3 4 5 6 7 8

Cf. A086270.

Cf. A000124, A050407, A001519. - Gary W. Adamson, Feb 18 2010

Row 1 = A000027. All subsequent rows are 0 followed by A000027.

A185780 Array T(n,k) = k(nk-n+1), by antidiagonals.

Original entry on oeis.org

1, 4, 1, 9, 6, 1, 16, 15, 8, 1, 25, 28, 21, 10, 1, 36, 45, 40, 27, 12, 1, 49, 66, 65, 52, 33, 14, 1, 64, 91, 96, 85, 64, 39, 16, 1, 81, 120, 133, 126, 105, 76, 45, 18, 1, 100, 153, 176, 175, 156, 125, 88, 51, 20, 1, 121, 190, 225, 232, 217, 186, 145, 100, 57, 22, 1, 144, 231, 280, 297, 288, 259, 216, 165, 112, 63, 24, 1, 169, 276, 341, 370, 369, 344, 301, 246, 185, 124, 69, 26, 1, 196, 325, 408, 451, 460, 441, 400, 343, 276, 205, 136, 75, 28, 1

Offset: 1

Clark Kimberling, Feb 03 2011

This is the accumulation array of A185781, the weight array of A185782, and second weight array of A185783. See A144112 for definitions of accumulation array and weight array.

			Northwest corner:
  1....4....9....16....25....36
  1....6....15...28....45....66
  1....8....21...40....65....96
  1....10...27...52....85....126

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185781, A185782, A185783.

(* This code yields arrays A185780, A185781, and A185782. *)
f[n_,0]:=0;f[0,k_]:=0;  (* Used to make weight array A185782 *)
f[n_,k_]:=k(n*k-n+1);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* this array *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc array of {f(n,k)} *)
FullSimplify[s[n,k]]
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* A185781 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* array A185782 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* seq A185782 *)

T(n,k) = k*(n*k - n + 1), k>=1, n>=1.

A185872 Accumulation array of the (odd,odd)-polka dot array A185868, by antidiagonals.

Original entry on oeis.org

1, 5, 7, 16, 24, 22, 38, 59, 65, 50, 75, 120, 141, 136, 95, 131, 215, 262, 274, 245, 161, 210, 352, 440, 480, 470, 400, 252, 316, 539, 687, 770, 790, 741, 609, 372, 453, 784, 1015, 1160, 1225, 1208, 1099, 880, 525, 625, 1095, 1436, 1666, 1795, 1825, 1750, 1556, 1221, 715, 836, 1480, 1962, 2304, 2520, 2616, 2590, 2432, 2124, 1640, 946, 1090, 1947, 2605, 3090, 3420, 3605, 3647, 3540

Offset: 1

Clark Kimberling, Feb 05 2011

See A144112 for the definition of accumulation array.

			Northwest corner:
   1,   5,  16,  38,  75
   7,  24,  59, 120, 215
  22,  54, 141, 262, 440
  50, 136, 174, 480, 770

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A185868.

Row 1: A174723; column 1: A002412.

f[n_,k_]:=2n-1+(2n+2k-4)(2n+2k-3)/2;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185868 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]] (*formula for A185872 *)
g[n_]:=Sum[f[n+1-k,k],{k,1,n}];
Table[g[n],{n,50}] (* A185872 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]

T(n,k) = (k*n/6)*(4*n^2 + 6*n*k + 4*k^2 - 3*n - 9*k + 4), k>=1, n>=1.

A185877 Array T given by T(n,k) = k^2 +(2n-3)k -2*n +3, by antidiagonals.

Original entry on oeis.org

1, 3, 1, 7, 5, 1, 13, 11, 7, 1, 21, 19, 15, 9, 1, 31, 29, 25, 19, 11, 1, 43, 41, 37, 31, 23, 13, 1, 57, 55, 51, 45, 37, 27, 15, 1, 73, 71, 67, 61, 53, 43, 31, 17, 1, 91, 89, 85, 79, 71, 61, 49, 35, 19, 1, 111, 109, 105, 99, 91, 81, 69, 55, 39, 21, 1, 133, 131, 127, 121, 113, 103, 91, 77, 61, 43, 23, 1, 157, 155, 151, 145, 137, 127, 115, 101, 85, 67, 47, 25, 1, 183, 181, 177, 171, 163, 153, 141, 127, 111, 93, 73, 51, 27, 1

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... (See A144112 for the definition of accumulation array).

			Northwest corner:
  1, 3,  7, 13, 21
  1, 5, 11, 19, 29
  1, 7, 15, 25, 45
  1, 9, 19, 31, 45

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185878, A185879, A185880.
Row 1 to 3: A002061, A028387, A082111.
diag (1,5,...): A056108;
diag (3,11,...): A056106;
diag (7,19,...): A003215;
diag (13,29,...): A144391;
diag (1,7,...): A003215;
diag (1,9,...): A144390.

(* This program generates A185877, its accumulation array A185878, and its weight array A185879. *)
f[n_,0]:=0;f[0,k_]:=0;
f[n_,k_]:=k^2+(2n-3)k-2n+3;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185877 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]]  (* formula for A185878 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A185879 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = k^2 + (2*n-3)*k - 2*n + 3, k>=1, n>=1.

A185878 Accumulation array of A185877, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 11, 10, 3, 24, 28, 18, 4, 45, 60, 51, 28, 5, 76, 110, 108, 80, 40, 6, 119, 182, 195, 168, 115, 54, 7, 176, 280, 318, 300, 240, 156, 70, 8, 249, 408, 483, 484, 425, 324, 203, 88, 9, 340, 570, 696, 728, 680, 570, 420, 256, 108, 10, 451, 770, 963, 1040, 1015, 906, 735, 528, 315, 130, 11, 584, 1012, 1290, 1428, 1440, 1344, 1162, 920, 648, 380, 154, 12

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ...

See A144112 for the definition of accumulation array.

			Northwest corner:
  1,  4, 11,  24,  45, ...
  2, 10, 28,  60, 110, ...
  3, 18, 51, 108, 195, ...
  4, 28, 80, 168, 300, ...
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185877, A185879.
Row 1 to 3: A006527, A006331, A064043.
Column 1 to 5: A000027, A028552, A140677, 12*A000096, 5*A130861.

f[n_, k_] := k*n*(2*k^2 - 3*k + 3*k*n - 3*n + 7)/6; Table[f[n - k + 1, k], {n,10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 21 2017 *)

T(n,k) = k*n*(2*k^2 -3*k +3*k*n -3*n +7)/6, k>=1, n>=1.

A185879 Weight array of A185877, by antidiagonals.

Original entry on oeis.org

1, 2, 0, 4, 2, 0, 6, 2, 2, 0, 8, 2, 2, 2, 0, 10, 2, 2, 2, 2, 0, 12, 2, 2, 2, 2, 2, 0, 14, 2, 2, 2, 2, 2, 2, 0, 16, 2, 2, 2, 2, 2, 2, 2, 0, 18, 2, 2, 2, 2, 2, 2, 2, 2, 0, 20, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 24, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 26, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ...< A185879 < A185877 < A185878 < A185880 <...

See A144112 for the definitions of weight array and accumulation array.

			Northwest corner:
1...2...4...6...8...10...12...14
0...2...2...2...2...2....2....2
0...2...2...2...2...2....2....2
0...2...2...2...2...2....2....2
0...2...2...2...2...2....2....2

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185877.

f[n_, k_] := 2; f[1, k_] := 2*(k - 1); f[n_, 1] := 0; f[1, 1] := 1;
TableForm[Table[f[n, k], {n, 1, 7}, {k, 1, 7}]] Table[f[n - k + 1, k], {n, 5}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 21 2017 *)

T(1,k) = 2*(k-1) for k>=1; T(n,1)=0 for n>1; T(n,k)=2 otherwise.

A185880 Second accumulation array of A185877, by antidiagonals.

Original entry on oeis.org

1, 5, 3, 16, 17, 6, 40, 56, 38, 10, 85, 140, 128, 70, 15, 161, 295, 320, 240, 115, 21, 280, 553, 670, 600, 400, 175, 28, 456, 952, 1246, 1250, 1000, 616, 252, 36, 705, 1536, 2128, 2310, 2075, 1540, 896, 348, 45, 1045, 2355, 3408, 3920, 3815, 3185, 2240, 1248, 465, 55, 1496, 3465, 5190, 6240, 6440, 5831, 4620, 3120, 1680, 605, 66, 2080, 4928, 7590, 9450, 10200, 9800, 8428, 6420, 4200, 2200

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... See A144112 for the definition of accumulation array.

			Northwest corner:
   1,    5,   16,   40,   85
   3,   17,   56,  140,  295
   6,   38,  128,  320,  670
  10,   70,  240,  600, 1250

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185877, A185878.
Antidiagonal sums: A037235.
diag (1,5,...): A056108 (4th spoke on hexagonal wheel);
diag (3,11,...): A056106 (2nd spoke on hexagonal wheel);
diag (7,19,...): A003215 (hex numbers);
diag (13,29,...): A144391.

(* This program generates A185878 first and then generates A185880 as the accumulation array of A185878. *)
f[n_,k_]:=(k*n/6)(7-3k+2k^2-3n+3kn);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185878 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];
FullSimplify[s[n,k]]
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185880 *)
f[n_, k_] := (1/72)*k*(1 + k)*n*(1 + n)*(16 - k + 3 *k^2 + 4 *(-1 + k) *n); Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 21 2017 *)

T(n,k) = C(k,2)*C(n,2)*(3*k^2+4*k*n-k-4*n+16)/18, k>=1, n>=1.

cp's OEIS Frontend

A185910 Array: T(n,k) = n^2 + k - 1, by antidiagonals.

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A185957 Second accumulation array of the array min{n,k}, by antidiagonals.

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A144148 Weight array W={w(i,j)} of the Wythoff array A035513.

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A144257 Weight array of A086270.

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A185780 Array T(n,k) = k*(n*k-n+1), by antidiagonals.

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A185872 Accumulation array of the (odd,odd)-polka dot array A185868, by antidiagonals.

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A185877 Array T given by T(n,k) = k^2 +(2*n-3)*k -2*n +3, by antidiagonals.

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A185878 Accumulation array of A185877, by antidiagonals.

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A185780 Array T(n,k) = k(nk-n+1), by antidiagonals.

A185877 Array T given by T(n,k) = k^2 +(2n-3)k -2*n +3, by antidiagonals.