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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.

%I A386625 #9 Aug 04 2025 06:25:59
%S A386625 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,
%T A386625 65,30,7,34,68,102,84,105,48,14,64,55,110,165,136,170,78,35,104,27,89,
%U A386625 178,267,220,275,126,56,168,45,50,144,288,432,356,445,204,91,272,72,80,11
%N 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.
%C A386625 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.
%F A386625 (row 1) = (1,2,3,5,8,13,...) = Fibonacci numbers, {F(n), n>=2}, and for m>1,
%F A386625 (row m) = numbers m*F(n) that are not h*F(k) for any h<m and k>=2.
%e A386625 Corner:
%e A386625    1    2    3    5    8   13   21   34   55
%e A386625    4    6   10   16   26   42   68  110  178
%e A386625    9   15   24   39   63  102  165  267  432
%e A386625   12   20   32   52   84  136  220  356  576
%e A386625   25   40   65  105  170  275  445  720 1165
%e A386625   18   30   48   78  126  204  330  534  864
%e A386625    7   14   35   56   91  147  238  385  623
%e A386625   64  104  168  272  440  712 1152  864 3016
%e A386625   27   45   72  117  189  306  495  801 1296
%e A386625   50   80  130  210  340  550  890 1440 2330
%e A386625   11   22   33   88  143  231  374  605  979
%e A386625   36   60   96  156  252  408  660 1068 1728
%e A386625 Row 4 is obtained from 4*(row 1) by removing 4 and 8.
%t A386625 ClearAll[rArray]
%t A386625 rArray[rows_, cols_] := Module[{fibs, R = {}, used = <||>, row, val, f, m},
%t A386625    fibs = Map[Fibonacci, Range[2, cols + rows]];
%t A386625    Table[row = {};
%t A386625     Do[val = m*f; If[! KeyExistsQ[used, val], AppendTo[row, val];
%t A386625       used[val] = True;], {f, fibs}]; Take[row, cols], {m, rows}]];
%t A386625 Grid[rArray[16, 12], Frame -> All]  (* array *)
%t A386625 r[m_, n_] := rArray[12, 12][[m]][[n]];
%t A386625 Table[r[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (*sequence*)
%t A386625 (* _Peter J. C. Moses_, Jul 27 2025 *)
%Y A386625 Cf. A000045, A005086 (counts removed duplicates), A035513, A386214.
%K A386625 nonn,tabl
%O A386625 1,2
%A A386625 _Clark Kimberling_, Jul 27 2025