A361992 -id:A361992 - cp's OEIS frontend

A361993 (2,1)-block array, B(2,1), of the Wythoff array (A035513), read by descending antidiagonals.

5, 9, 15, 14, 25, 26, 23, 40, 43, 36, 37, 65, 69, 59, 47, 60, 105, 112, 95, 77, 57, 97, 170, 181, 154, 124, 93, 68, 157, 275, 293, 249, 201, 150, 111, 78, 254, 445, 474, 403, 325, 243, 179, 127, 89, 411, 720, 767, 652, 526, 393, 290, 205, 145, 99, 665, 1165

Offset: 1

Clark Kimberling, Apr 04 2023

We begin with a definition. Suppose that W = (w(i,j)), where i >= 1 and j >= 1, is an array of numbers such that if m and n satisfy 1 <= m < n, then there exists k such that w(m,k+h) < w(n,h+1) < w(m,k+h+1) for every h >= 0. Then W is a row-splitting array. The array B(2,1) is a row-splitting array. The rows of B(2,1) are linearly recurrent with signature (1,1); the columns are linearly recurrent with signature (1,1,-1). The order array (as defined in A333029) of B(2,1) is A361995.

			Corner of B(2,1):
   5    9   14   23   37   60   97  157 ...
  15   25   40   65  105  170  275  445 ...
  26   43   69  112  181  293  474  767 ...
  36   59   95  154  249  403  652 1055 ...
  47   77  124  202  325  526  851 1377 ...
  ...
(column 1 of A035513) = (1,4,6,9,12,14,17,19,...), so (column 1 of B(2,1)) = (5,15,26,36,...);
(column 2 of A000027) = (2,7,10,15,20,23,28,31,...), so (column 2 of B(2,1)) = (9,25,43,59,...).

Cf. A000045, A001622, A035513, A080164, A361975, A361992 (array B(1,2)), A361994 (array B(2,2)).

f[n_] := Fibonacci[n]; r = GoldenRatio;
zz = 10; z = 13;
w[n_, k_] := f[k + 1] Floor[n*r] + (n - 1) f[k]
t[h_, k_] := w[2 h - 1, k] + w[2 h, k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A361993 sequence *)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* A361993 array *)

B(2,1) = (b(i,j)), where b(i,j) = w(2i-1,j) + w(2i,j) for i >= 1, j >= 1, where (w(i,j)) is the Wythoff array (A035513).

b(i,j) = F(j+1) ([2 i r] + [(2 i - 1) r]) + (4 i - 3) F(j), where F = A000045, the Fibonacci numbers, and r = (1+sqrt(5))/2, the golden ratio, A001622, and [ ] = floor.

A361994 (2,2)-block array, B(2,1), of the Wythoff array (A035513), read by descending antidiagonals.

Original entry on oeis.org

14, 37, 40, 97, 105, 69, 254, 275, 181, 95, 665, 720, 474, 249, 124, 1741, 1885, 1241, 652, 325, 150, 4558, 4935, 3249, 1707, 851, 393, 179, 11933, 12920, 8506, 4469, 2228, 1029, 469, 205, 31241, 33825, 22269, 11700, 5833, 2694, 1228, 537, 234, 81790, 88555

Offset: 1

Clark Kimberling, Apr 04 2023

We begin with a definition. Suppose that W = (w(i,j)), where i >= 1 and j >= 1, is an array of numbers such that if m and n satisfy 1 <= m < n, then there exists k such that w(m,k+h) < w(n,h+1) < w(m,k+h+1) for every h >= 0. Then W is a row-splitting array. The array B(2,2) is a row-splitting array. The rows of B(2,2) are linearly recurrent with signature (3,-1); the columns are linearly recurrent with signature (1,1,-1). The order array (as defined in A333029) of B(2,2) is A361996.

			Corner of B(2,2):
   14    37    97   254   665   1741 ...
   40   105   275   720  1885   4935 ...
   69   181   474  1241  3249   8506 ...
   95   249   652  1707  4469  11700 ...
  124   325   851  2228  5833  15271 ...
  ...
b(1,1) = w(1,1) + w(1,2) + w(2,1) + w(2,2) = 1 +  2 +  4 +  7 = 14;
b(1,2) = w(1,3) + w(1,4) + w(2,3) + w(2,4) = 3 +  5 + 11 + 18 = 37;
b(2,1) = w(3,1) + w(3,2) + w(4,1) + w(4,2) = 8 + 10 +  9 + 15 = 40.

Cf. A000045, A001622, A035513, A080164, A361976, A361992 (array B(1,2)), A361993 (array B(2,1)).

f[n_] := Fibonacci[n]; r = GoldenRatio;
zz = 10; z = 13;
w[n_, k_] := f[k + 1] Floor[n*r] + (n - 1) f[k]
t[h_, k_] := w[2 h - 1, 2 k - 1] + w[2 h - 1, 2 k] + w[2 h, 2 k - 1] + w[2 h, 2 k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (*A361994 sequence *)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* A361994 array *)

B(2,2) = (b(i,j)), where b(i,j) = w(2i-1,2j-1) + w(2i-1,2j) + w(2i,2j-1) + w(2i,2j) for i >= 1, j >= 1, where (w(i,j)) is the Wythoff array (A035513).

cp's OEIS Frontend

A361993 (2,1)-block array, B(2,1), of the Wythoff array (A035513), read by descending antidiagonals.

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