A185812 -id:A185812 - 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.

A185912 Accumulation array of A185910; by antidiagonals.

Original entry on oeis.org

1, 3, 5, 6, 12, 14, 10, 21, 31, 30, 15, 32, 51, 64, 55, 21, 45, 74, 102, 115, 91, 28, 60, 100, 144, 180, 188, 140, 36, 77, 129, 190, 250, 291, 287, 204, 45, 96, 161, 240, 325, 400, 441, 416, 285, 55, 117, 196, 294, 405, 515, 602, 636, 579, 385, 66, 140, 234, 352, 490, 636, 770, 864, 882, 780, 506, 78, 165, 275, 414, 580, 763, 945, 1100, 1194, 1185, 1023, 650, 91, 192, 319, 480, 675, 896, 1127, 1344, 1515, 1600, 1551, 1312, 819, 105

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185910 < A185911 < A185912 < A185913 < ...

(See A144112 for definitions of weight array and accumulation array.)

			Northwest corner:
   1,   3,   6,  10,  15
   5,  12,  21,  32,  45
  14,  31,  51,  74, 100
  30,  64, 102, 144, 190

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

Cf. A144112, A185910, A185913.
Row 1 to 2: A000217, A028347.
Column 1 to 3: A000330, A037237, 3*A145066.

f[n_, 0] := 0; f[0, k_] := 0;
f[n_, k_] := n^2 + k - 1;
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*)
Table[s[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten
T[n_, k_] := (k*n/6)*(2*n^2 + 3*n + 3*k - 2) ; Table[T[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 22 2017 *)

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

cp's OEIS Frontend

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

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