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

1,2

Comments

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

Links

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

Formula

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

Example

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

Mathematica

(* 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

Crossrefs

Cf. A144112, A185910, A185912.

Sequence in context: A246162 A243573 A132193 * A306779 A349947 A351652

Adjacent sequences: A185907 A185908 A185909 * A185911 A185912 A185913

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 06 2011

Status

approved