A185780 Array T(n,k) = k*(n*k-n+1), by antidiagonals.
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
Examples
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
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Programs
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Mathematica
(* 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 *)
Formula
T(n,k) = k*(n*k - n + 1), k>=1, n>=1.
Comments