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A185737 Accumulation array of the Wythoff array, by antidiagonals. 1

%I #18 Feb 26 2023 03:38:02

%S 1,3,5,6,14,11,11,28,30,20,19,51,60,54,32,32,88,109,108,86,46,53,148,

%T 188,196,172,123,63,87,245,316,338,312,246,168,82,142,402,523,568,538,

%U 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 Accumulation array of the Wythoff array, by antidiagonals.

%C For the definition of accumulation array, see A144112.

%e Northwest corner:

%e 1 3 6 11 19 (A001911)

%e 5 14 28 51 88

%e 11 30 60 109 188

%e 20 54 108 196 338

%t (* This program creates the Wythoff array W={f(n,k)}=A035513, then the accumulation array A185736 of W, then the weight array A185736 of W *)

%t f[n_,0]:=0;f[0,k_]:=0; (* Needed for the weight array *)

%t f[n_,k_]:=Fibonacci[k+1]Floor[n*GoldenRatio]+(n-1)Fibonacci[k];

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

%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}];

%t TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185736 *)

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

%t (* 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 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}]] (* A185736 *)

%t Table[w[n-k+1,k],{n,20},{k,n,1,-1}]//Flatten

%o (PARI) W(n, k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513

%o T(n, k) = sum(i=1, n, sum(j=1, k, W(i, j))); \\ _Michel Marcus_, Feb 25 2023

%Y Cf. A144112, A035513, A185736.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 02 2011

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