A194059 Natural interspersion of A001911 (Fibonacci numbers minus 2); a rectangular array, by antidiagonals.

1, 3, 2, 6, 4, 5, 11, 7, 8, 9, 19, 12, 13, 14, 10, 32, 20, 21, 22, 15, 16, 53, 33, 34, 35, 23, 24, 17, 87, 54, 55, 56, 36, 37, 25, 18, 142, 88, 89, 90, 57, 58, 38, 26, 27, 231, 143, 144, 145, 91, 92, 59, 39, 40, 28, 375, 232, 233, 234, 146, 147, 93, 60, 61, 41, 29

Offset: 1

Author

Clark Kimberling, Aug 14 2011

Keywords

• nonn
• tabl

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194059 is a permutation of the positive integers; its inverse is A194060.

Examples

			Northwest corner:
1...3...6...11...19
2...4...7...12...30
5...8...13..21...34
9...14..22..35...56
10..15..23..36...57
		

Crossrefs

Cf. A194029, A194059, A194062.

Programs

Mathematica

z = 50;
c[k_] := -2 + Fibonacci[k + 3];
c = Table[c[k], {k, 1, z}]  (* A001911, F(n+3)-2 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 700}]   (* cf. A194055 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A194059 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 100}]] (* A194060 *)