A340244 Wythoff-B array read by antidiagonals.

4, 7, 12, 11, 20, 17, 18, 32, 28, 25, 29, 52, 45, 41, 33, 47, 84, 73, 66, 54, 38, 76, 136, 118, 107, 87, 62, 46, 123, 220, 191, 173, 141, 100, 75, 51, 199, 356, 309, 280, 228, 162, 121, 83, 59, 322, 576, 500, 453, 369, 262, 196, 134, 96, 67, 521, 932, 809

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..58.

Formula

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.)

Example

Corner:
   4    7    11    18    29    47   76     123    199
  12   20    32    52    84   136   220    356    576
  17   28    45    73   118   191   309    500    809
  25   41    66   107   173   280   453    733   1186
  33   54    87   141   228   369   597    966   1563
  38   62   100   162   262   424   686   1110   1796
  46   75   121   196   317   513   830   1343   2173
  51   83   134   217   351   568   919   1487   2406

Mathematica

r = GoldenRatio; f[n_] := Fibonacci[n];
a[n_] := Floor[r*n]; b[n_] := Floor[r^2*n];
c[n_] := a[a[b[n]]]; d[n_] := b[a[b[n]]];
w[n_, k_] := f[k - 2] c[n] + f[k - 1] d[n];
Grid[Table[w[n, k], {n, 1, 15}, {k, 1, 15}]] (* A340244 array *)
Table[w[n - k + 1, k], {n, 15}, {k, n, 1, -1}] // Flatten  (* A340244 sequence *)

Crossrefs

Cf. A000045, A000201, A001950, A134859, A340245.

Sequence in context: A083487 A253129 A249918 * A158937 A092572 A344159

Adjacent sequences: A340241 A340242 A340243 * A340245 A340246 A340247

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 02 2021

Status

approved