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A185737 Accumulation array of the Wythoff array, read by antidiagonals.

%I A185737 #22 Feb 15 2025 17:12:21
%S A185737 1,3,5,6,14,11,11,28,30,20,19,51,60,54,32,32,88,109,108,86,46,53,148,
%T A185737 188,196,172,123,63,87,245,316,338,312,246,168,82,142,402,523,568,538,
%U A185737 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
%N A185737 Accumulation array of the Wythoff array, read by antidiagonals.
%C A185737 For the definition of accumulation array, see A144112.
%e A185737 Northwest corner:
%e A185737    1    3    6   11   19  (A001911)
%e A185737    5   14   28   51   88
%e A185737   11   30   60  109  188
%e A185737   20   54  108  196  338
%t A185737 (* 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 *)
%t A185737 f[n_,0]:=0;f[0,k_]:=0;  (* Needed for the weight array *)
%t A185737 f[n_,k_]:=Fibonacci[k+1]Floor[n*GoldenRatio]+(n-1)Fibonacci[k];
%t A185737 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* Wythoff array *)
%t A185737 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185737 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];
%t A185737 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A144148 *)
%t A185737 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185737 (* 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. *)
%t A185737 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 A185737 TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A144148 *)
%t A185737 Table[w[n-k+1,k],{n,20},{k,n,1,-1}]//Flatten
%o A185737 (PARI) W(n, k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513
%o A185737 T(n, k) = sum(i=1, n, sum(j=1, k, W(i, j))); \\ _Michel Marcus_, Feb 25 2023
%Y A185737 Cf. A144112, A035513, A144148.
%K A185737 nonn,tabl
%O A185737 1,2
%A A185737 _Clark Kimberling_, Feb 02 2011