A194030 Natural interspersion of the Fibonacci sequence (1,2,3,5,8,...), a rectangular array, by antidiagonals.

1, 2, 4, 3, 6, 7, 5, 9, 10, 11, 8, 14, 15, 16, 12, 13, 22, 23, 24, 17, 18, 21, 35, 36, 37, 25, 26, 19, 34, 56, 57, 58, 38, 39, 27, 20, 55, 90, 91, 92, 59, 60, 40, 28, 29, 89, 145, 146, 147, 93, 94, 61, 41, 42, 30, 144, 234, 235, 236, 148, 149, 95, 62, 63, 43, 31

Offset: 1

Author

Clark Kimberling, Aug 12 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, this sequence is a permutation of the positive integers; its inverse is A194031.

Examples

			Northwest corner:
  1...2...3...5...8...13
  4...6...9...14..22..35
  7...10..15..23..36..57
  11..16..24..37..58..92
  12..17..25..38..59..93
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened).
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A130233, A194029, A194031 (inverse).

Column 1 appears to be A345347.

Although A383977 initially has many terms in comment with this, the author of A383977 has observed that the two sequences are unrelated. - N. J. A. Sloane, May 17 2025

Programs

Mathematica

z = 40;
c[k_] := Fibonacci[k + 1];
c = Table[c[k], {k, 1, z}]  (* A000045 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194029 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A194030 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194031 *)
(* Second program: *)
f[x_] := Block[{i = 1}, While[x > Fibonacci[i], i++]; i - 1]; Table[Fibonacci[f[k] + n - k + 1] + k - 1, {n, 2, 12}, {k, n - 1}] // Flatten (* Michael De Vlieger, May 17 2025 *)

Formula

T(n,k) = Fibonacci(A130233(n) + k - 1) + n - 1. - Michael De Vlieger, May 17 2025