A194061 Natural interspersion of A002620; a rectangular array, by antidiagonals.

1, 2, 3, 4, 5, 8, 6, 7, 11, 15, 9, 10, 14, 19, 24, 12, 13, 18, 23, 29, 35, 16, 17, 22, 28, 34, 41, 48, 20, 21, 27, 33, 40, 47, 55, 63, 25, 26, 32, 39, 46, 54, 62, 71, 80, 30, 31, 38, 45, 53, 61, 70, 79, 89, 99, 36, 37, 44, 52, 60, 69, 78, 88, 98, 109, 120

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, A194061 is a permutation of the positive integers; its inverse is A194062.

Examples

			Northwest corner:
1...2...4...6...9...12
3...5...7...10..13..17
8...11..14..18..22..27
15..19..23..28..33..39
24..29..34..40..46..53
		

Crossrefs

Cf. A194029, A002620, A122197, A194062.

Programs

Mathematica

z = 50;
c[k_] := Floor[((k + 1)^2)/4];
c = Table[c[k], {k, 1, z}]  (* A002620 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 400}]   (* [A122197] *)
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, 11}, {k, 1, n}]] (* A194061 *)
q[n_] := Position[p, n]; Flatten[
 Table[q[n], {n, 1, 90}]]  (* A194062 *)