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A340244 Wythoff-B array read by antidiagonals.

%I A340244 #5 Jan 05 2021 14:44:41
%S A340244 4,7,12,11,20,17,18,32,28,25,29,52,45,41,33,47,84,73,66,54,38,76,136,
%T A340244 118,107,87,62,46,123,220,191,173,141,100,75,51,199,356,309,280,228,
%U A340244 162,121,83,59,322,576,500,453,369,262,196,134,96,67,521,932,809
%N A340244 Wythoff-B array read by antidiagonals.
%C A340244 The Wythoff array, A134859, consists of columns AA, BA, ABA, BBA, ABBA, BBBA, ...  The Wythoff-B array consists of columns AAB, BAB, ABAB, BBAB, ABBAB, BBBAB, ... , formed by suffixing B to the column designations for A134859.  Column k shows the numbers whose Zeckendorf representation has least terms F(k+1) and F(k+2), where F = A000045, the Fibonacci numbers. The rows are interspersed, in the sense that the order array (A340245) of the Wythoff-B array is an interspersion.
%F A340244 For n >=1 and k >= 1, w(n,k) = F(k-2)*A(A(B(n))) + F(k-1)*B(A(B(n))), where A(n) = floor(n*phi), B(n) = floor(n*phi^2); i.e., A = A000201, B= A001950, these being the lower and upper Wythoff sequences. (Note that F(-1) = 1, F(0) = 0.)
%e A340244 Corner:
%e A340244    4    7    11    18    29    47   76     123    199
%e A340244   12   20    32    52    84   136   220    356    576
%e A340244   17   28    45    73   118   191   309    500    809
%e A340244   25   41    66   107   173   280   453    733   1186
%e A340244   33   54    87   141   228   369   597    966   1563
%e A340244   38   62   100   162   262   424   686   1110   1796
%e A340244   46   75   121   196   317   513   830   1343   2173
%e A340244   51   83   134   217   351   568   919   1487   2406
%t A340244 r = GoldenRatio; f[n_] := Fibonacci[n];
%t A340244 a[n_] := Floor[r*n]; b[n_] := Floor[r^2*n];
%t A340244 c[n_] := a[a[b[n]]]; d[n_] := b[a[b[n]]];
%t A340244 w[n_, k_] := f[k - 2] c[n] + f[k - 1] d[n];
%t A340244 Grid[Table[w[n, k], {n, 1, 15}, {k, 1, 15}]] (* A340244 array *)
%t A340244 Table[w[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten  (* A340244 sequence *)
%Y A340244 Cf. A000045, A000201, A001950, A134859, A340245.
%K A340244 nonn,tabl
%O A340244 1,1
%A A340244 _Clark Kimberling_, Jan 02 2021