A185737 Accumulation array of the Wythoff array, read by antidiagonals.
1, 3, 5, 6, 14, 11, 11, 28, 30, 20, 19, 51, 60, 54, 32, 32, 88, 109, 108, 86, 46, 53, 148, 188, 196, 172, 123, 63, 87, 245, 316, 338, 312, 246, 168, 82, 142, 402, 523, 568, 538, 446, 336, 218, 104, 231, 656, 858, 940, 904, 769, 609, 436, 276, 129, 375, 1067, 1400, 1542, 1496, 1292, 1050, 790, 552, 342, 156, 608, 1732, 2277, 2516, 2454, 2138, 1764, 1362, 1000, 684, 413, 186
Offset: 1
Examples
Northwest corner: 1 3 6 11 19 (A001911) 5 14 28 51 88 11 30 60 109 188 20 54 108 196 338
Programs
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Mathematica
(* This program creates the Wythoff array W={f(n,k)} = A035513, then the accumulation array A185737 of W, then the weight array A144148 of W *) f[n_,0]:=0;f[0,k_]:=0; (* Needed for the weight array *) f[n_,k_]:=Fibonacci[k+1]Floor[n*GoldenRatio]+(n-1)Fibonacci[k]; TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* Wythoff 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}]; TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A144148 *) Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* In general,the weight array W of an arbitrary rectangular array S={s(i,j):i<=1,j<=1} is defined in two steps:(1) define s(i,j)=0 if i=0 or j=0; (2) then w(m,n)=s(m,n)+s(m-1,n-1)-s(m,n-1)-s(m-1,n) for m<1,n<1. *) 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}]] (* A144148 *) Table[w[n-k+1,k],{n,20},{k,n,1,-1}]//Flatten -
PARI
W(n, k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513 T(n, k) = sum(i=1, n, sum(j=1, k, W(i, j))); \\ Michel Marcus, Feb 25 2023
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