A185910 - OEIS

%I #24 Oct 25 2021 08:20:48

%S 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,

%T 49,8,10,14,20,28,38,50,64,9,11,15,21,29,39,51,65,81,10,12,16,22,30,

%U 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

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

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

%H G. C. Greubel, <a href="/A185910/b185910.txt">Table of n, a(n) for the first 50 antidiagonals, flattened</a>

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

%e Northwest corner:

%e    1,  2,  3,  4,  5

%e    4,  5,  6,  7,  8

%e    9, 10, 11, 12, 13

%e   16, 17, 18, 19, 20

%t (* This program generates the array A185910, its accumulation array A185812, and its weight array A185911. *)

%t f[n_,0]:=0;f[0,k_]:=0;

%t f[n_,k_]:=n^2+k-1;

%t TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185910 *)

%t Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

%t s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)

%t FullSimplify[s[n,k]]  (* formula for A185812 *)

%t TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]

%t Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

%t w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];

%t TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A185911 *)

%t Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

%Y Cf. A144112, A185910, A185912.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 06 2011