%I #5 Jan 05 2021 14:44:41
%S 4,7,12,11,20,17,18,32,28,25,29,52,45,41,33,47,84,73,66,54,38,76,136,
%T 118,107,87,62,46,123,220,191,173,141,100,75,51,199,356,309,280,228,
%U 162,121,83,59,322,576,500,453,369,262,196,134,96,67,521,932,809
%N Wythoff-B array read by antidiagonals.
%C 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 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 Corner:
%e 4 7 11 18 29 47 76 123 199
%e 12 20 32 52 84 136 220 356 576
%e 17 28 45 73 118 191 309 500 809
%e 25 41 66 107 173 280 453 733 1186
%e 33 54 87 141 228 369 597 966 1563
%e 38 62 100 162 262 424 686 1110 1796
%e 46 75 121 196 317 513 830 1343 2173
%e 51 83 134 217 351 568 919 1487 2406
%t r = GoldenRatio; f[n_] := Fibonacci[n];
%t a[n_] := Floor[r*n]; b[n_] := Floor[r^2*n];
%t c[n_] := a[a[b[n]]]; d[n_] := b[a[b[n]]];
%t w[n_, k_] := f[k - 2] c[n] + f[k - 1] d[n];
%t Grid[Table[w[n, k], {n, 1, 15}, {k, 1, 15}]] (* A340244 array *)
%t Table[w[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten (* A340244 sequence *)
%Y Cf. A000045, A000201, A001950, A134859, A340245.
%K nonn,tabl
%O 1,1
%A _Clark Kimberling_, Jan 02 2021