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 A194029 Natural fractal sequence of the Fibonacci sequence (1,2,3,5,8,...). 37
 1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Suppose that c(1), c(2), c(3),... is a strictly increasing sequence of positive integers with c(1)=1, and that the sequence c(k+1)-c(k) is strictly increasing.  The natural fractal sequence f of c is here introduced by the following rule: ... If c(k)<=n= 1 (see example). - Omar E. Pol, May 28 2012 REFERENCES Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168. LINKS Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117. EXAMPLE The sequence (1,2,3,5,8,13,...) is used to place 1's in positions numbered 1,2,3,5,8,13,...  Then gaps are filled in with consecutive counting numbers:   1,1,1,2,1,2,3,1,2,3,4,5,1,... From Omar E. Pol, May 28 2012: (Start) Written as an irregular triangle the sequence begins: 1; 1; 1,2; 1,2,3; 1,2,3,4,5; 1,2,3,4,5,6,7,8; 1,2,3,4,5,6,7,8,9,10,11,12,13; 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21; The row lengths are A000045(n). (End) MATHEMATICA z = 40; c[k_] := Fibonacci[k + 1]; c = Table[c[k], {k, 1, z}]  (* A000045 *) f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]] f = Table[f[n], {n, 1, 800}]  (* A194029 *) 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, 13}, {k, 1, n}]]  (* A194030 *) q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194031 *) Flatten[Range[Fibonacci[Range[66]]]] (* Birkas Gyorgy, Jun 30 2012 *) CROSSREFS Cf. A000045, A194030, A194031. Sequence in context: A194844 A138528 A037125 * A194055 A162192 A238954 Adjacent sequences:  A194026 A194027 A194028 * A194030 A194031 A194032 KEYWORD nonn,tabf AUTHOR Clark Kimberling, Aug 12 2011 STATUS approved

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Last modified August 21 19:20 EDT 2018. Contains 313955 sequences. (Running on oeis4.)