A185778 -id:A185778 - cp's OEIS frontend

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

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_{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.

			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

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

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 *)

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

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.

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

A185779 Third accumulation array of Pascal's triangle (as a rectangle), by antidiagonals.

Original entry on oeis.org

1, 4, 4, 10, 17, 10, 20, 45, 45, 20, 35, 95, 126, 95, 35, 56, 175, 281, 281, 175, 56, 84, 294, 546, 662, 546, 294, 84, 120, 462, 966, 1358, 1358, 966, 462, 120, 165, 690, 1596, 2534, 2941, 2534, 1596, 690, 165, 220, 990, 2502, 4410, 5790, 5790, 4410, 2502, 990, 220, 286, 1375, 3762, 7272, 10620, 12021, 10620, 7272, 3762, 1375, 286, 364, 1859, 5467, 11484, 18432, 23229, 23229, 18432, 11484, 5467, 1859, 364, 455

Offset: 1

Clark Kimberling, Feb 03 2011

Using "Axxxxxx < Ayyyyyy" to mean that Ayyyyyy is the accumulation array of Axxxxxx, as defined at A144112:
A185779 < A144225 < A007318 < A014430 < A077023 < A185779, where each of these is formatted as a rectangle (e.g., A007318 is Pascal's triangle).  See A185778.
row 1: A000292
row 2: A095667

			Northwest corner:
1....4...10...20...35
4....17..45...95...175
10...45..126..281..546
20...95..281..662..1358

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A007318, A014430, A077023, A144112, A144225, A185778.

f[n_, k_] := Binomial[n + k + 4, n + 2] - (k + 3)*(k + 4)/2 - (k + 2)* n*(k*n + n + 3*k + 7)/4; TableForm[Table[f[n, k], {n, 1, 5}, {k, 1, 5}]]
Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 12 2017 *)

T(n,k) = C(n+k+4,n+2) - (k+3)*(k+4)/2 - (k+2)*n*(k*n+n+3*k+7)/4, for k>=1, n>=1.

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

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