A272904 -id:A272904 - cp's OEIS frontend

A272900 Fibonacci-products fractal sequence.

1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 4, 2, 1, 3, 4, 2, 1, 3, 5, 4, 2, 1, 3, 5, 4, 2, 1, 3, 5, 6, 4, 2, 1, 3, 5, 6, 4, 2, 1, 3, 5, 7, 6, 4, 2, 1, 3, 5, 7, 6, 4, 2, 1, 3, 5, 7, 8, 6, 4, 2, 1, 3, 5, 7, 8, 6, 4, 2, 1, 3, 5, 7, 9, 8, 6, 4, 2, 1, 3, 5, 7, 9

Offset: 1

Clark Kimberling, May 10 2016

Let F = A000045, the Fibonacci numbers. Let s be the sequence of all products F(i)F(j), for 2 <= i < = j, arranged in increasing order; viz., (1,2,3,4,5,6,8,9,10,13,15,...) = (F(2)F(2), F(2)F(3), F(2)F(4), F(3)F(3), F(2)F(5), ... ), as at A049997. The sequence of first factors is (F(2), F(2), F(2), F(3), F(2),...), represented by indices (2,2,2,3,2,...). Subtracting 1 from each term leaves A272900, which is a fractal sequence; i.e., the removal of the first occurrence of each term in A272900 leaves A272900, so that the sequence contains itself infinitely many times.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35.

Cf. A272904 (the associated interspersion), A000045, A049997, A272907 (Lucas-products fractal sequence).

z = 200; f[n_] := Fibonacci[n + 1]; u1 = Table[f[n], {n, 1, z}];
u2 = Sort[Flatten[Table[f[i]*f[j], {i, 1, z}, {j, i, z}]]];
Table[Select[Range[30], MemberQ[u1, u2[[i]]/f[#]] &][[1]], {i, 1, z}]

A272908 Rectangular array, by antidiagonals: row n give the positions of n in the Lucas-products fractal sequence, A272907.

Original entry on oeis.org

1, 2, 5, 3, 7, 8, 4, 10, 11, 16, 6, 13, 14, 20, 23, 9, 17, 18, 25, 28, 35, 12, 21, 22, 30, 33, 41, 46, 15, 26, 27, 36, 39, 48, 53, 62, 19, 31, 32, 42, 45, 55, 60, 70, 77, 24, 37, 38, 49, 52, 63, 68, 79, 86, 97, 29, 43, 44, 56, 59, 71, 76, 88, 95, 107, 116

Offset: 1

Clark Kimberling, May 10 2016

This array is an interspersion. Every positive integer occurs exactly once, and each row is interspersed by each other row, except for initial terms.

			Northwest corner:
1     2     3     4     6     9     12    15
5     7     10    13    17    21    26    31
8     11    14    18    22    27    32    38
16    20    25    30    36    42    49    56
23    28    33    39    45    52    59    67
35    41    48    55    63    71    80    89
46    53    60    68    76    85    94    104

Cf. A000032, A272907, A272909, A272904 (Fibonacci-products interspersion).

z = 500; f[n_] := LucasL[n]; u1 = Table[f[n], {n, 1, z}];
u2 = Sort[Flatten[Table[f[i]*f[j], {i, 1, z}, {j, i, z}]]];
uf = Table[Select[Range[80], MemberQ[u1, u2[[i]]/f[#]] &][[1]], {i, 1, z}]
r[n_, k_] := Flatten[Position[uf, n]][[k]]
TableForm[Table[r[n, k], {n, 1, 12}, {k, 1, 12}]]  (* A272908 array *)
Table[r[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (* A272908 sequence *)

cp's OEIS Frontend

A272900 Fibonacci-products fractal sequence.

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