A194038 Natural interspersion of A034856, a rectangular array, by antidiagonals.

1, 4, 2, 8, 5, 3, 13, 9, 6, 7, 19, 14, 10, 11, 12, 26, 20, 15, 16, 17, 18, 34, 27, 21, 22, 23, 24, 25, 43, 35, 28, 29, 30, 31, 32, 33, 53, 44, 36, 37, 38, 39, 40, 41, 42, 64, 54, 45, 46, 47, 48, 49, 50, 51, 52, 76, 65, 55, 56, 57, 58, 59, 60, 61, 62, 63, 89, 77, 66

Offset: 0

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, A194038 is a permutation of the positive integers; its inverse is A194040.

Examples

			Northwest corner:
1...4...8...13...19
2...5...9...14...20
3...6...10..15...21
7...11..16..22...29
12..17..23..30...38
		

Crossrefs

Cf. A192872, A194040.

Programs

Mathematica

z = 30;
c[k_] := (k^2 + 3 k - 2)/2;
c = Table[c[k], {k, 1, z}]  (* A034856 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 255}]  (* essentially A002260 *)
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, 13}, {k, 1, n}]]   (* A194038 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194040 *)