A194108 Natural interspersion of A194106; a rectangular array, by antidiagonals.

1, 4, 2, 9, 5, 3, 15, 10, 6, 7, 23, 16, 11, 12, 8, 33, 24, 17, 18, 13, 14, 45, 34, 25, 26, 19, 20, 21, 58, 46, 35, 36, 27, 28, 29, 22, 73, 59, 47, 48, 37, 38, 39, 30, 31, 90, 74, 60, 61, 49, 50, 51, 40, 41, 32, 109, 91, 75, 76, 62, 63, 64, 52, 53, 42, 43, 129, 110

Offset: 1

Author

Clark Kimberling, Aug 15 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, A194108 is a permutation of the positive integers; its inverse is A194109.

Examples

			Northwest corner:
1...4...9...15...23
2...5...10..16...24
3...6...11..17...25
7...12..18..26...36
8...13..19..27...37
		

Crossrefs

Cf. A194029, A194106, A194107, A194109.

Programs

Mathematica

z = 40; g = Sqrt[3];
c[k_] := Sum[Floor[j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A194106 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194107 *)
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, 16}, {k, 1, n}]]  (* A194108 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194109 *)