cp's OEIS Frontend

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

%I A361994 #10 Apr 07 2023 17:38:05
%S A361994 14,37,40,97,105,69,254,275,181,95,665,720,474,249,124,1741,1885,1241,
%T A361994 652,325,150,4558,4935,3249,1707,851,393,179,11933,12920,8506,4469,
%U A361994 2228,1029,469,205,31241,33825,22269,11700,5833,2694,1228,537,234,81790,88555
%N A361994 (2,2)-block array, B(2,1), of the Wythoff array (A035513), read by descending antidiagonals.
%C A361994 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.
%F A361994 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).
%e A361994 Corner of B(2,2):
%e A361994    14    37    97   254   665   1741 ...
%e A361994    40   105   275   720  1885   4935 ...
%e A361994    69   181   474  1241  3249   8506 ...
%e A361994    95   249   652  1707  4469  11700 ...
%e A361994   124   325   851  2228  5833  15271 ...
%e A361994   ...
%e A361994 b(1,1) = w(1,1) + w(1,2) + w(2,1) + w(2,2) = 1 +  2 +  4 +  7 = 14;
%e A361994 b(1,2) = w(1,3) + w(1,4) + w(2,3) + w(2,4) = 3 +  5 + 11 + 18 = 37;
%e A361994 b(2,1) = w(3,1) + w(3,2) + w(4,1) + w(4,2) = 8 + 10 +  9 + 15 = 40.
%t A361994 f[n_] := Fibonacci[n]; r = GoldenRatio;
%t A361994 zz = 10; z = 13;
%t A361994 w[n_, k_] := f[k + 1] Floor[n*r] + (n - 1) f[k]
%t A361994 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];
%t A361994 Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (*A361994 sequence *)
%t A361994 TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* A361994 array *)
%Y A361994 Cf. A000045, A001622, A035513, A080164, A361976, A361992 (array B(1,2)), A361993 (array B(2,1)).
%K A361994 nonn,tabl
%O A361994 1,1
%A A361994 _Clark Kimberling_, Apr 04 2023