This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A194029 #67 Dec 11 2024 15:32:15
%S A194029 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,
%T A194029 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,
%U A194029 7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34
%N A194029 Natural fractal sequence of the Fibonacci sequence (1, 2, 3, 5, 8, ...).
%C A194029 Suppose that c(1), c(2), c(3), ... is an 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 defined by:
%C A194029 If c(k) <= n < c(k+1), then f(n) = 1 + n - c(k).
%C A194029 This defines the present sequence a(n) = f(n) for c = A000045.
%C A194029 The natural interspersion of c is here introduced as the array given by T(n,k) =(position of k-th n in f). Note that c = (row 1 of T).
%C A194029 As a different example from the one considered here (c = A000045), let c = A000217 = (1, 3, 6, 10, 15, ...), the triangular numbers, so that f = (1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, ...) = A002260, and a northwest corner of T = A194029 is:
%C A194029 1 3 6 10 15 ...
%C A194029 2 4 7 11 16 ...
%C A194029 5 8 12 17 23 ...
%C A194029 9 13 18 24 31 ...
%C A194029 ...
%C A194029 Since every number in the set N of positive integers occurs exactly once in this (and every) interspersion, a listing of the terms of T by antidiagonals comprises a permutation, p, of N; letting q denote the inverse of p, we thus have for each c a fractal sequence, an interspersion T, and two permutations of N:
%C A194029 c f T / p q
%C A194029 A000045 A194029 A194030 A194031
%C A194029 A000290 A071797 A194032 A194033
%C A194029 A000217 A002260 A066182 A066181
%C A194029 A028387 A074294 A194034 A194035
%C A194029 A028872 A071797 A194036 A194037
%C A194029 A034856 A002260 A194038 A194040
%C A194029 It appears that this is also a triangle read by rows in which row n lists the first A000045(n) positive integers, n >= 1 (see example). - _Omar E. Pol_, May 28 2012
%C A194029 This is true, because the sequence c = A000045 has the property that c(k+1) - c(k) = c(k-1), so the number of integers {1, 2, 3, ...} to be filled in from index n = c(k) to n = c(k+1)-1 is equal to c(k-1); see also the first EXAMPLE. - _M. F. Hasler_, Apr 23 2022
%D A194029 Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.
%H A194029 Alois P. Heinz, <a href="/A194029/b194029.txt">Rows n = 1..20, flattened</a>
%H A194029 Clark Kimberling, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa73/aa7321.pdf">Numeration systems and fractal sequences</a>, Acta Arithmetica 73 (1995) 103-117.
%F A194029 a(n) = A066628(n)+1. - _Alan Michael Gómez Calderón_, Oct 30 2023
%e A194029 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:
%e A194029 1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, ...
%e A194029 From _Omar E. Pol_, May 28 2012: (Start)
%e A194029 Written as an irregular triangle the sequence begins:
%e A194029 1;
%e A194029 1;
%e A194029 1, 2;
%e A194029 1, 2, 3;
%e A194029 1, 2, 3, 4, 5;
%e A194029 1, 2, 3, 4, 5, 6, 7, 8;
%e A194029 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13;
%e A194029 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21; ...
%e A194029 The row lengths are A000045(n).
%e A194029 (End)
%p A194029 T:= n-> $1..(<<0|1>, <1|1>>^n)[1, 2]:
%p A194029 seq(T(n), n=1..10); # _Alois P. Heinz_, Dec 11 2024
%t A194029 z = 40;
%t A194029 c[k_] := Fibonacci[k + 1];
%t A194029 c = Table[c[k], {k, 1, z}] (* A000045 *)
%t A194029 f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
%t A194029 f = Table[f[n], {n, 1, 800}] (* A194029 *)
%t A194029 r[n_] := Flatten[Position[f, n]]
%t A194029 t[n_, k_] := r[n][[k]]
%t A194029 TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
%t A194029 p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]] (* A194030 *)
%t A194029 q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]] (* A194031 *)
%t A194029 Flatten[Range[Fibonacci[Range[66]]]] (* _Birkas Gyorgy_, Jun 30 2012 *)
%Y A194029 Cf. A000045 (Fibonacci numbers).
%Y A194029 Cf. A066628, A194030, A194031 (natural interspersion of A000045 and inverse permutation).
%Y A194029 Cf. A130853.
%K A194029 nonn,tabf
%O A194029 1,4
%A A194029 _Clark Kimberling_, Aug 12 2011
%E A194029 Edited by _M. F. Hasler_, Apr 23 2022