A194034 Natural interspersion of A028387, a rectangular array, by antidiagonals.

1, 5, 2, 11, 6, 3, 19, 12, 7, 4, 29, 20, 13, 8, 9, 41, 30, 21, 14, 15, 10, 55, 42, 31, 22, 23, 16, 17, 71, 56, 43, 32, 33, 24, 25, 18, 89, 72, 57, 44, 45, 34, 35, 26, 27, 109, 90, 73, 58, 59, 46, 47, 36, 37, 28, 131, 110, 91, 74, 75, 60, 61, 48, 49, 38, 39, 155, 132

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, A194034 is a permutation of the positive integers; its inverse is A194035.

Examples

			Northwest corner:
1...5...11...19...29...41
2...6...12...20...30...42
3...7...13...21...31...43
4...8...14...22...32...44
9...15..23...33...45...59
		

Crossrefs

Cf. A194029, A194035.

Programs

Mathematica

z = 30;
c[k_] := k^2 + k - 1;
c = Table[c[k], {k, 1, z}]  (* A028387 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 255}]  (* A074294 *)
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}]]  (* A194034 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194035 *)