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 A087088 #62 Sep 26 2022 01:59:01
%S A087088 1,2,3,1,4,2,1,5,3,1,2,6,1,4,2,1,3,7,1,2,5,1,3,2,1,4,8,1,2,3,1,6,2,1,
%T A087088 4,3,1,2,5,1,9,2,1,3,4,1,2,7,1,3,2,1,5,4,1,2,3,1,6,2,1,10,3,1,2,4,1,5,
%U A087088 2,1,3,8,1,2,4,1,3,2,1,6,5,1,2,3,1,4,2,1,7,3,1,2,11,1,4,2,1,3,5,1,2,6,1,3,2
%N A087088 Positive ruler-type fractal sequence with 1's in every third position.
%C A087088 If all the terms in the sequence are reduced by one and then all zeros are removed, the result is the same as the original sequence.
%C A087088 From _Benoit Cloitre_, Mar 07 2009: (Start)
%C A087088 To construct the sequence:
%C A087088 Step 1: start from a sequence of 1's and leave two undefined places between every pair of 1's giving: 1,(),(),1,(),(),1,(),(),1,(),(),1,(),(),1,...
%C A087088 Step 2: replace the first undefined place with a 2 and henceforth leave two undefined places between two 2's giving: 1,2,(),1,(),2,1,(),(),1,2,(),1,(),2,1,...
%C A087088 Step 3: replace the first undefined place with a 3 and henceforth leave two undefined places between two 3's giving: 1,2,3,1,(),2,1,(),3,1,2,(),1,(),2,1,...
%C A087088 Step 4: replace the first undefined place with a 4 and leave 2 undefined places between two 4's giving: 1,2,3,1,4,2,1,(),3,1,2,(),1,4,2,1,... Iterating the process indefinitely yields the sequence: 1,2,3,1,4,2,1,5,3,1,2,6,1,4,2,1,... (End)
%C A087088 From _Peter Munn_, Jul 10 2020: (Start)
%C A087088 For k >= 1, the number k occurs in a pattern with fundamental period 3^k, and with points of mirror symmetry at intervals of (3^k)/2. Those points have an extrapolated common origin (for k >= 1) at an offset 1.5 to the left of the sequence's initial "1". The snake format illustration in the example section may be useful for visualizing this.
%C A087088 (End)
%C A087088 For k >= 1, k first occurs at position A061419(k) and its k-th occurrence is at position A083045(k-1). - _Peter Munn_, Aug 23 2020
%C A087088 (a(n)) is the unique fixed point of the two-block substitution a,b -> 1,a+1,b+1, where a,b are natural numbers. - _Michel Dekking_, Sep 26 2022
%H A087088 Alois P. Heinz, <a href="/A087088/b087088.txt">Table of n, a(n) for n = 1..20000</a>
%F A087088 a(n) = 1 when n == 1 (mod 3), otherwise a(n) = a(n-ceiling(n/3)) + 1.
%F A087088 a(n) = 3 + A244040(3*(n-1)) - A244040(3*n). - _Tom Edgar_ and _James Van Alstine_, Aug 04 2014
%F A087088 From _Peter Munn_, Aug 22 2020: (Start)
%F A087088 For m >= 0, a(3*m+1) = 1; a(3*m+2) = a(2*m+1) + 1; a(3*m+3) = a(2*m+2) + 1.
%F A087088 For n >= 1, the following identities hold.
%F A087088 a(n) = A335933(2*n+1).
%F A087088 A083044(A163491(n) - 1, a(n) - 1) = n.
%F A087088 A051064(n+1) = min(a(n), a(n+1)).
%F A087088 A254046(n+2) = min(a(n), a(n+2)). (End)
%e A087088 From _Peter Munn_, Jul 03 2020: (Start)
%e A087088 Listing the terms in a snake format (with period 27) illustrates periodic and mirror symmetries. Horizontal lines mark points of mirror symmetry for 3's. Vertical lines mark further points of mirror symmetry for 2's. 79 terms are shown. (Referred to the extrapolated common origin of periodic mirror symmetry, the initial term is at offset 1.5 and the last shown is at offset 79.5 = 3^4 - 1.5.) Observe also mirror symmetry of 4's (seen vertically).
%e A087088 1 2 3 1 4 2 1 5 3 1 2 6
%e A087088 | | 1 --
%e A087088 1 2 3 1 5 2 1 7 3 1 2 4
%e A087088 _ 4
%e A087088 8
%e A087088 1 2 3 1 6 2 1 4 3 1 2 5
%e A087088 | | 1 --
%e A087088 1 2 3 1 7 2 1 4 3 1 2 9
%e A087088 _ 5
%e A087088 4
%e A087088 1 2 3 1 6 2 1 10 3 1 2 4
%e A087088 | | 1 --
%e A087088 1 2 3 1 4 2 1 8 3 1 2 5
%e A087088 (End)
%e A087088 From _Peter Munn_, Aug 22 2020: (Start)
%e A087088 The start of the sequence is shown below in conjunction with related sequences, aligning their points of mirror symmetry. The longer, and shorter, vertical lines indicate points of mirror symmetry for terms valued less than 4, and less than 3, respectively. Note each term of A051064 is the minimum of two terms displayed nearest below it, and each term of A254046 is the minimum of the two terms displayed diagonally above it.
%e A087088 | | |
%e A087088 A051064:| 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1 4 1 1 2
%e A087088 | | | | | | |
%e A087088 [a(n)]: | 1 2 3 1 4 2 1 5 3 1 2 6 1 4 2 1 3 7 1 2 5 1 3 2 1 4 8 1 2 3
%e A087088 | | | | | | |
%e A087088 A254046:|1 2 1 1 3 1 1 2 1 1 2 1 1 4 1 1 2 1 1 2 1 1 3 1 1 2 1 1 2 1 1
%e A087088 | | |
%e A087088 (End)
%t A087088 A163491[n_] := A163491[n] = If[Mod[n, 3]==1, (n+2)/3, A163491[Floor[2n/3]]];
%t A087088 Module[{b}, b[_] = 0;
%t A087088 a[n_] := With[{t = A163491[n]}, b[t] = b[t] + 1]];
%t A087088 Array[a, 105] (* _Jean-François Alcover_, Jan 10 2022 *)
%Y A087088 Cf. A001511, A244040.
%Y A087088 Sequences with equivalent symmetries: A051064, A254046.
%Y A087088 Records are given by A061419: a(A061419(n))=n.
%Y A087088 Essentially the odd bisection of A335933.
%Y A087088 Sequence with similar definition: A087165.
%Y A087088 Ordinal transform of A163491, with which this sequence has a joint relationship to A083044, A083045.
%Y A087088 See also the comment in A024629.
%K A087088 nonn,easy,look
%O A087088 1,2
%A A087088 Enrico T. Federighi (rico125162(AT)aol.com), Aug 08 2003
%E A087088 More terms from _Paul D. Hanna_, Aug 21 2003
%E A087088 Offset changed by _M. F. Hasler_ (following remarks by _Peter Munn_), Jul 13 2020
%E A087088 Thanks to _Allan C. Wechsler_ for suggesting the new name. - _N. J. A. Sloane_, Jul 14 2020