A386625 Rectangular array R, read by descending antidiagonals: (row m) = increasing sequence of numbers m*F(n), n>=2, without duplicates, where F = A000045 (Fibonacci numbers); see Comments.

1, 2, 4, 3, 6, 9, 5, 10, 15, 12, 8, 16, 24, 20, 25, 13, 26, 39, 32, 40, 18, 21, 42, 63, 52, 65, 30, 7, 34, 68, 102, 84, 105, 48, 14, 64, 55, 110, 165, 136, 170, 78, 35, 104, 27, 89, 178, 267, 220, 275, 126, 56, 168, 45, 50, 144, 288, 432, 356, 445, 204, 91, 272, 72, 80, 11

Offset

1,2

Comments

Every positive integer appears exactly once. Every non-Fibonacci prime appears in the first column. Except for initial terms, every row is a row of the Wythoff array, A035513.

Links

Table of n, a(n) for n=1..66.

Formula

(row 1) = (1,2,3,5,8,13,...) = Fibonacci numbers, {F(n), n>=2}, and for m>1,

(row m) = numbers m*F(n) that are not h*F(k) for any h<m and k>=2.

Example

Corner:
   1    2    3    5    8   13   21   34   55
   4    6   10   16   26   42   68  110  178
   9   15   24   39   63  102  165  267  432
  12   20   32   52   84  136  220  356  576
  25   40   65  105  170  275  445  720 1165
  18   30   48   78  126  204  330  534  864
   7   14   35   56   91  147  238  385  623
  64  104  168  272  440  712 1152  864 3016
  27   45   72  117  189  306  495  801 1296
  50   80  130  210  340  550  890 1440 2330
  11   22   33   88  143  231  374  605  979
  36   60   96  156  252  408  660 1068 1728
Row 4 is obtained from 4*(row 1) by removing 4 and 8.

Mathematica

ClearAll[rArray]
rArray[rows_, cols_] := Module[{fibs, R = {}, used = <||>, row, val, f, m},
   fibs = Map[Fibonacci, Range[2, cols + rows]];
   Table[row = {};
    Do[val = m*f; If[! KeyExistsQ[used, val], AppendTo[row, val];
      used[val] = True; ], {f, fibs}]; Take[row, cols], {m, rows}]];
Grid[rArray[16, 12], Frame -> All]  (* array *)
r[m_, n_] := rArray[12, 12][[m]][[n]];
Table[r[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (*sequence*)
(* Peter J. C. Moses, Jul 27 2025 *)

Crossrefs

Cf. A000045, A005086 (counts removed duplicates), A035513, A386214.

Sequence in context: A389088 A359299 A363473 * A361748 A377093 A372337

Adjacent sequences: A386622 A386623 A386624 * A386626 A386627 A386628

Keyword

nonn,tabl

Author

Clark Kimberling, Jul 27 2025

Status

approved