OFFSET
1,2
COMMENTS
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 06 2011
STATUS
approved