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A087088
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Positive ruler-type fractal sequence with 1's in every third position.
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10
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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, 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, 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
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OFFSET
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1,2
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COMMENTS
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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.
To construct the sequence:
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,...
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,...
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,...
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)
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.
(End)
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
(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
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LINKS
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FORMULA
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a(n) = 1 when n == 1 (mod 3), otherwise a(n) = a(n-ceiling(n/3)) + 1.
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.
For n >= 1, the following identities hold.
A254046(n+2) = min(a(n), a(n+2)). (End)
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EXAMPLE
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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).
1 2 3 1 4 2 1 5 3 1 2 6
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1 2 3 1 5 2 1 7 3 1 2 4
_ 4
8
1 2 3 1 6 2 1 4 3 1 2 5
| | 1 --
1 2 3 1 7 2 1 4 3 1 2 9
_ 5
4
1 2 3 1 6 2 1 10 3 1 2 4
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1 2 3 1 4 2 1 8 3 1 2 5
(End)
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.
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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
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[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
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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
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(End)
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MATHEMATICA
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Module[{b}, b[_] = 0;
a[n_] := With[{t = A163491[n]}, b[t] = b[t] + 1]];
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CROSSREFS
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Essentially the odd bisection of A335933.
Sequence with similar definition: A087165.
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KEYWORD
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AUTHOR
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Enrico T. Federighi (rico125162(AT)aol.com), Aug 08 2003
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EXTENSIONS
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STATUS
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approved
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