A328696 Rectangular array R read by descending antidiagonals: apply x -> (x+1)/2 to each odd term of the Wythoff array (A035513), and delete all others.

1, 2, 4, 3, 6, 5, 7, 15, 8, 12, 11, 24, 20, 19, 9, 28, 62, 32, 49, 23, 10, 45, 100, 83, 79, 37, 16, 13, 117, 261, 134, 206, 96, 41, 21, 14, 189, 422, 350, 333, 155, 66, 54, 36, 25, 494, 1104, 566, 871, 405, 172, 87, 58, 40, 17, 799, 1786, 1481, 1409, 655

Offset: 1

Author

Clark Kimberling, Oct 26 2019

Keywords

• nonn
• tabl

Comments

Every positive integer occurs exactly once in R, and every row of R is a linear recurrence sequence.

Examples

			Row 1 of the Wythoff array is (1,2,3,5,8,13,21,34,55,89,144,...), so that row 1 of R is (1,2,3,7,11,...) = A107857 (essentially).
_______________
Northwest corner of R:
   1,  2,  3,  7,  11,  28,  45,  117,  189,  494,   799
   4,  6, 15, 24,  62, 100, 261,  422, 1104, 1786,  4675
   5,  8, 20, 32,  83, 134, 350,  566, 1481, 2396,  6272
  12, 19, 49, 79, 206, 333, 871, 1409, 3688, 5967, 15621
   9, 23, 37, 96, 155, 405, 655, 1714, 2773, 7259, 11745
  10, 16, 41, 66, 172, 278, 727, 1176, 3078, 4980, 13037
  13, 21, 54, 87, 227, 367, 960, 1553, 4065, 6577, 17218
		

Crossrefs

Cf. A035513, A107857, A328695, A328697.

Programs

Mathematica

w[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten;
q[n_, k_] := If[Mod[w[n, k], 2] == 1, (1 + w[n, k])/2, 0];
t[n_] := Union[Table[q[n, k], {k, 1, 50}]];
u[n_] := If[First[t[n]] == 0, Rest[t[n]], t[n]]
s = Select[Range[40], ! u[#] == {} &]; u1[n_] := u[s[[n]]];
Column[Table[u1[n], {n, 1, 10}]] (* A328696 array *)
v[n_, k_] := u1[n][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A328696 sequence *)