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

%I A361993 #11 Jun 04 2023 23:50:26
%S A361993 5,9,15,14,25,26,23,40,43,36,37,65,69,59,47,60,105,112,95,77,57,97,
%T A361993 170,181,154,124,93,68,157,275,293,249,201,150,111,78,254,445,474,403,
%U A361993 325,243,179,127,89,411,720,767,652,526,393,290,205,145,99,665,1165
%N A361993 (2,1)-block array, B(2,1), of the Wythoff array (A035513), read by descending antidiagonals.
%C A361993 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.
%F A361993 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).
%F A361993 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.
%e A361993 Corner of B(2,1):
%e A361993    5    9   14   23   37   60   97  157 ...
%e A361993   15   25   40   65  105  170  275  445 ...
%e A361993   26   43   69  112  181  293  474  767 ...
%e A361993   36   59   95  154  249  403  652 1055 ...
%e A361993   47   77  124  202  325  526  851 1377 ...
%e A361993   ...
%e A361993 (column 1 of A035513) = (1,4,6,9,12,14,17,19,...), so (column 1 of B(2,1)) = (5,15,26,36,...);
%e A361993 (column 2 of A000027) = (2,7,10,15,20,23,28,31,...), so (column 2 of B(2,1)) = (9,25,43,59,...).
%t A361993 f[n_] := Fibonacci[n]; r = GoldenRatio;
%t A361993 zz = 10; z = 13;
%t A361993 w[n_, k_] := f[k + 1] Floor[n*r] + (n - 1) f[k]
%t A361993 t[h_, k_] := w[2 h - 1, k] + w[2 h, k];
%t A361993 Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A361993 sequence *)
%t A361993 TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* A361993 array *)
%Y A361993 Cf. A000045, A001622, A035513, A080164, A361975, A361992 (array B(1,2)), A361994 (array B(2,2)).
%K A361993 nonn,tabl
%O A361993 1,1
%A A361993 _Clark Kimberling_, Apr 04 2023