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

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

Author

Clark Kimberling, Apr 04 2023

Keywords

• nonn
• tabl

Comments

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.

Examples

			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.
		

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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).