%I #13 Mar 30 2012 18:57:38

%S 1,2,3,4,5,1,2,3,4,5,6,7,1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10,11,

%T 12,13,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,1,2,3,4,5,6,7,8,9,10,11,12,

%U 13,14,15,16,17,18,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17

%N Natural fractal sequence of A194126.

%C See A194029 for definitions of natural fractal sequence and natural interspersion.

%t z = 40; g = GoldenRatio;

%t c[k_] := -1 + Sum[Floor[j + j*g], {j, 1, k}];

%t c = Table[c[k], {k, 1, z}] (* A194126 *)

%t f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]

%t f = Table[f[n], {n, 1, 800}] (* A193042 *)

%t r[n_] := Flatten[Position[f, n]]

%t t[n_, k_] := r[n][[k]]

%t TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]

%t p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]] (* A194100 *)

%t q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]] (* A194101 *)

%Y Cf. A194029, A194126, A194100.

%K nonn

%O 1,2

%A _Clark Kimberling_, Aug 15 2011