A194032 Natural interspersion of the squares (1,4,9,16,25,...), a rectangular array, by antidiagonals.

1, 4, 2, 9, 5, 3, 16, 10, 6, 7, 25, 17, 11, 12, 8, 36, 26, 18, 19, 13, 14, 49, 37, 27, 28, 20, 21, 15, 64, 50, 38, 39, 29, 30, 22, 23, 81, 65, 51, 52, 40, 41, 31, 32, 24, 100, 82, 66, 67, 53, 54, 42, 43, 33, 34, 121, 101, 83, 84, 68, 69, 55, 56, 44, 45

Offset: 1

Author

Clark Kimberling, Aug 12 2011

Keywords

• nonn
• easy
• 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, A194032 is a permutation of the positive integers; its inverse is A194033.

Examples

			Northwest corner:
  1...4...9...16...25
  2...5...10..17...26
  3...6...11..18...27
  7...12..19..28...39
  8...13..20..29...40
		

Links

• Zhuorui He, Table of n, a(n) for n = 1..11325

Crossrefs

Cf. A000290, A071797, A194033, A192872.

Programs

Mathematica

z = 30;
c[k_] := k^2;
c = Table[c[k], {k, 1, z}]  (* A000290 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]] (* A071797 *)
f = Table[f[n], {n, 1, 255}]
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, 14}, {k, 1, n}]] (* A194032 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]] (* A194033 *)

Formula

T(n, k) = (k + max(floor(n/2)-1,0))^2 + n - 1. - Zhuorui He, Jul 08 2025