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

1,2

Comments

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

Links

Table of n, a(n) for n=1..60.

Example

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).
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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

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 *)

Crossrefs

Cf. A035513, A107857, A328695, A328697.

Sequence in context: A366949 A332072 A330340 * A053629 A227370 A135761

Adjacent sequences: A328693 A328694 A328695 * A328697 A328698 A328699

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 26 2019

Status

approved