A144148 -id:A144148 - cp's OEIS frontend

A185736 Duplicate of A144148.

1, 1, 3, 1, 2, 2, 2, 3, 1, 3, 3, 5, 2, 2, 3, 5, 8, 3, 3, 2, 2, 8, 13, 5, 5, 3, 1, 3, 13, 21, 8, 8, 5, 2, 2, 2, 21, 34, 13, 13, 8, 3, 3, 1, 3, 34, 55, 21, 21, 13, 5, 5, 2, 2, 3, 55, 89, 34, 34, 21, 8, 8, 3, 3, 2, 2, 89, 144, 55, 55, 34, 13, 13, 5, 5, 3, 1

Offset: 1

dead

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

Original entry on oeis.org

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

Clark Kimberling, Feb 02 2011

For the definition of accumulation array, see A144112.

			Northwest corner:
   1    3    6   11   19  (A001911)
   5   14   28   51   88
  11   30   60  109  188
  20   54  108  196  338

Cf. A144112, A035513, A144148.

(* 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

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

A144149 Weight array W={w(i,j)} of the Wythoff difference array A080164.

Original entry on oeis.org

1, 1, 2, 3, 3, 1, 8, 8, 2, 2, 21, 21, 5, 3, 2, 55, 55, 13, 8, 3, 1, 144, 144, 34, 21, 8, 2, 2, 377, 377, 89, 55, 21, 5, 3, 1, 987, 987, 233, 144, 55, 13, 8, 2, 2, 2584, 2584, 610, 377, 144, 34, 21, 5, 3, 2, 6765, 6765, 1597, 987, 377, 89, 55, 13, 8, 3, 1, 17711, 17711, 4181

Offset: 1

Clark Kimberling, Sep 11 2008

In general, let w(i,j) be the weight of the unit square labeled by its northeast vertex (i,j) and for each (m,n), define
  S(m,n) = Sum_{j=1..n} Sum_{i=1..m} w(i,j).
Then S(m,n) is the weight of the rectangle [0,m]x[0,n]. We call W the weight array of S and we call S the accumulation array of W. For the case at hand, S is the Wythoff difference array, A080164.

			S(2,4) = 1+1+3+8+2+3+8+21 = 47.

Cf. A000045, A144112, A144148.

Row 1: 1 followed by A001906, except for initial 0.
Row n: A001519 (except for initial 1) if n is in 1+A001950.
Row n: A001906 (except for initial 0) if n is in 1+A000201.

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A185736 Duplicate of A144148.

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

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A144149 Weight array W={w(i,j)} of the Wythoff difference array A080164.

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