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A272904 Rectangular array, by antidiagonals: row n gives the positions of n in the Fibonacci-products fractal sequence, A272900.

%I A272904 #10 Jan 05 2025 19:51:40
%S A272904 1,2,4,3,6,8,5,9,11,15,7,12,14,19,23,10,16,18,24,28,34,13,20,22,29,33,
%T A272904 40,46,17,25,27,35,39,47,53,61,21,30,32,41,45,54,60,69,77,26,36,38,48,
%U A272904 52,62,68,78,86,96,31,42,44,55,59,70,76,87,95,106,116
%N A272904 Rectangular array, by antidiagonals: row n gives the positions of n in the Fibonacci-products fractal sequence, A272900.
%C A272904 This array is an interspersion.  Every positive integer occurs exactly once, and each row is interspersed by each other row, except for initial terms.
%C A272904 Row 1:  A033638  (quarter-squares plus 1)
%C A272904 Row 2:  A002620  (quarter-squares)
%C A272904 Column 1:  A267682 (conjectured)
%H A272904 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-1/quartkimberling01_2004.pdf">Orderings of products of Fibonacci numbers</a>, Fibonacci Quarterly 42:1 (2004), pp. 28-35.
%e A272904 Northwest corner:
%e A272904 1     2     3     4     6     9     12    15
%e A272904 5     7     10    13    17    21    26    31
%e A272904 8     11    14    18    2     27    32    38
%e A272904 16    20    25    30    36    42    49    56
%e A272904 23    28    33    39    45    52    59    67
%e A272904 35    41    48    55    63    71    80    89
%e A272904 46    53    60    68    76    85    94    104
%t A272904 z = 500; f[n_] := Fibonacci[n + 1]; u1 = Table[f[n], {n, 1, z}];
%t A272904 u2 = Sort[Flatten[Table[f[i]*f[j], {i, 1, z}, {j, i, z}]]];
%t A272904 uf = Table[Select[Range[80], MemberQ[u1, u2[[i]]/f[#]] &][[1]], {i, 1, z}]
%t A272904 r[n_, k_] := Flatten[Position[uf, n]][[k]]
%t A272904 TableForm[Table[r[n, k], {n, 1, 12}, {k, 1, 12}]]  (* A272904 array *)
%t A272904 t = Table[r[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A272904 sequence *)
%Y A272904 Cf. A000045, A272900, A033638, A002620, A267682, A272908 (Lucas-products interspersion).
%K A272904 nonn,tabl,easy
%O A272904 1,2
%A A272904 _Clark Kimberling_, May 10 2016