A328695 -id:A328695 - cp's OEIS frontend

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

Clark Kimberling, Oct 26 2019

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

			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

Cf. A035513, A107857, A328695, A328697.

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 *)

A328697 Rectangular array R read by descending antidiagonals: divide the multiples of 3 in the Wythoff array (A035513) by 3, and delete all others.

Original entry on oeis.org

1, 7, 6, 48, 41, 2, 329, 281, 14, 3, 2255, 1926, 96, 5, 4, 15456, 13201, 658, 8, 28, 20, 105937, 90481, 4510, 13, 192, 137, 15, 726103, 620166, 30912, 21, 1316, 939, 103, 27, 4976784, 4250681, 211874, 34, 9020, 6436, 706, 185, 12, 34111385, 29134601, 1452206

Offset: 1

Clark Kimberling, Oct 29 2019

Every positive integer occurs in R exactly once, and every row of R is a linear recurrence sequence.
Row 1 of R is essentially A004187.
Row 2 of R is essentially A049685.
Row 4 of R is essentially A000045.

			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 (1,7,48,329,2255,...).
=====================
Northwest corner of R:
   1,   7,  48,  329,  2255,  15456,  105937,   726103
   6,  41, 281, 1926, 13201,  90481,  620166,  4250681
   2,  14,  96,  658,  4510,  30912,  211874,  1452206
   3,   5,   8,   13,    21,     34,      55,       89
   4,  28, 192, 1316,  9020,  61824,  423748,  2904412
  20, 137, 939, 6436, 44113, 302355, 2072372, 14204249
  15, 103, 706, 4839, 33167, 227330, 1558143, 10679671

Cf. A035513, A004187, A049685, A000045, A328695, A328696.

cp's OEIS Frontend

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.

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