A144148 Weight array W={w(i,j)} of the Wythoff array A035513.

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, 3, 144, 233, 89, 89, 55, 21, 21, 8, 8, 5, 2, 2, 3

Offset

1,3

Comments

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_{i=1..m} Sum_{j=1..n} w(i,j).

Then S(m,n) is the weight of the rectangle [0,m]x[0,n]. As in A144112, we call W the weight array of S, and S the accumulation array of W, which can be derived from S as follows:

(1) extend S by defining S(i,j)=0 if i=0 or j=0; and

(2) then w(m,n) = s(m,n) + s(m-n,n-1) - s(m,n-1) - s(n,m-1) for m>=1, n>=1.

For the case at hand, S is the Wythoff array, A035513. These arrays form a chain:

... ->A144148->A035513->A185737-> ... Every term of this array is a Fibonacci number.

Links

Table of n, a(n) for n=1..91.

Formula

For m>3, if the row number is m of form floor(h*r+1), where r=(1+sqrt(5))/2, then

(row m)=(row 2); otherwise, (row m)=(row 3).

row n: (3,2,3,5,8,13,21,...) if n>1 is in the lower Wythoff sequence, A000201.

row n: (2,1,2,3,5,8,13,21,...) if n is in the upper Wythoff sequence, A001950.

Example

Corner:
    1  1  1  2  3   5   8  13  21  34   55   89
    3  2  3  5  8  13  21  34  55  89  144  233
    2  1  2  3  5   8  13  21  34  55   89  144
    3  2  3  5  8  13  21  34  55  89  144  233
    3  2  3  5  8  13  21  34  55  89  144  233
    2  1  2  3  5   8  13  21  34  55   89  144
    3  2  3  5  8  13  21  34  55  89  144  233
    2  1  2  3  5   8  13  21  34  55   89  144
    3  2  3  5  8  13  21  34  55  89  144  233
    3  2  3  5  8  13  21  34  55  89  144  233
    2  1  2  3  5   8  13  21  34  55   89  144
    3  2  3  5  8  13  21  34  55  89  144  233

Mathematica

s[n_, k_] := Fibonacci[k + 1]  Floor[n*GoldenRatio] + (n - 1)  Fibonacci[k];
Grid[Table[s[n, k], {n, 1, 12}, {k, 1, 12}]]   (* A035513 *)
s[0, k_] := 0; s[n_, 0] = 0;
w[m_, n_] := s[m, n] + s[m - 1, n - 1] - s[m, n - 1] - s[m - 1, n];
Grid[Table[w[n, k], {n, 1, 12}, {k, 1, 12}]] (* array *)
Table[w[k, m - k], {m, 2, 14}, {k, 1, m - 1}] // Flatten (* sequence *)

Prog

(PARI) s(n, k) = if ((n<=0) || (k<=0), 0, (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k)); \\ A035513
w(n, k) = s(n, k)+s(n-1, k-1)-s(n, k-1)-s(n-1, k); \\ Michel Marcus, Feb 02 2025

Crossrefs

Cf. A000045, A000201, A001950, A035513, A144112, A185737, A185778.

Sequence in context: A033178 A029418 A185736 * A343950 A085247 A003016

Adjacent sequences: A144145 A144146 A144147 * A144149 A144150 A144151

Keyword

nonn,tabl

Author

Clark Kimberling, Sep 11 2008

Extensions

Corrected and extended by Michel Marcus, Feb 02 2025

Some of the content of the duplicate (and now dead) sequence A185736 has been merged into this entry. - N. J. A. Sloane, Feb 15 2025

Edited by Clark Kimberling, Feb 16 2025

Status

approved