A194968 Fractalization of (1+[n/r]), where [ ]=floor, r=(1+sqrt(5))/2 (the golden ratio), and n>=1.

1, 1, 2, 1, 3, 2, 1, 3, 4, 2, 1, 3, 4, 5, 2, 1, 3, 4, 6, 5, 2, 1, 3, 4, 6, 7, 5, 2, 1, 3, 4, 6, 8, 7, 5, 2, 1, 3, 4, 6, 8, 9, 7, 5, 2, 1, 3, 4, 6, 8, 9, 10, 7, 5, 2, 1, 3, 4, 6, 8, 9, 11, 10, 7, 5, 2, 1, 3, 4, 6, 8, 9, 11, 12, 10, 7, 5, 2, 1, 3, 4, 6, 8, 9, 11, 12, 13, 10, 7, 5, 2, 1, 3, 4

Offset: 1

Author

Clark Kimberling, Sep 07 2011

Keywords

• nonn

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (1+[n/r]) is A019446.

Crossrefs

Cf. A194959, A019446, A194969, A194970.

Programs

Mathematica

r = GoldenRatio; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A019446 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]  (* A194968 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]]  (* A194969 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194970 *)