A185877 Array T given by T(n,k) = k^2 +(2*n-3)*k -2*n +3, by antidiagonals.
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
Examples
Northwest corner: 1, 3, 7, 13, 21 1, 5, 11, 19, 29 1, 7, 15, 25, 45 1, 9, 19, 31, 45
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Crossrefs
Programs
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Mathematica
(* 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
Formula
T(n,k) = k^2 + (2*n-3)*k - 2*n + 3, k>=1, n>=1.
Comments