cp's OEIS Frontend

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.

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A132827 Table based upon insertion points of n into sequence A132828 and having a specific formula.

Original entry on oeis.org

1, 3, 2, 8, 6, 4, 21, 16, 11, 5, 55, 42, 29, 14, 7, 144, 110, 76, 37, 19, 9, 377, 288, 199, 97, 50, 24, 10, 987, 754, 521, 254, 131, 63, 27, 12
Offset: 0

Views

Author

Kenneth J Ramsey, Sep 03 2007

Keywords

Comments

The numbers n in column j of this table always have (F(2j) -1) numbers less than n that appear before n in the sequence. For instance, 8 has 7 terms to the left thereof in the sequence that are less than 8, so 8 appears in column 3 of the table. Each positive integer has a unique position in the table.
This array was not known until after sequence A132828 was generated based upon the infinite Fibonacci word A005614 wherein the consecutive numbers 1 to 255 were inserted into the sequence being created at an insertion point based in part on the relative value of the infinite word after truncating the first n-1 terms.
The above rectangular array was generated by placing n into column j where j was the insertion point of n into the sequence. It was discovered that the insertion points were always 1,3,8,21,55,... counting from the left. I was trying to pick insertion points such that the value of the truncated Fibonacci word was always increasing but think I had an error in the program.
The array omits the empty columns. It appears the terms of other sequences can be uniquely placed into columns of a table by virtue of the number of terms to the left of each number in the array that are less than or equal to the number. For j > 3, A(0,j) = A(1,j-1) + A(1,j-2) - A(0,j-3); A(1,j) = A(2,j-1) + A(2,j-2) + A(1,j-3) - A(0,j-4).
Conjecture: The array A132827 is the dispersion of the sequence f given by f(n)=floor(n*x+n+1), where x=(golden ratio). Evidence: use f(n_):=Floor[n*x+n+1] in the Mathematica program at A191426. - Clark Kimberling, Jun 03 2011

Examples

			a(3,2) = (b(3)+1)*F(2*2) + (3 - b(3))*F(2*2+1). b(3) = 2 in A005206 so a(3,2)= 3*3 + 1*5 = 14.
Corner of the array:
  1,  3,  8, 21,  55
  2,  6, 16, 42, 110
  4, 11, 29, 76, 199
  5, 14, 37, 97, 254

Crossrefs

Cf. A191426.

Programs

  • Mathematica
    (See Conjecture under Comments.)

Formula

A(i,j) = (b(i)+1) * F(2j) + (i-b(i))*F(2j+1) where F(j) is the j-th Fibonacci number and b(n) = the n-th term of the Hofstadier G-sequence A005206.

A132917 Order set of the first 300 infinite truncated Fibonacci Words where a(n) is the number of terms (ones and zeros) truncated from the left hand side of the word.

Original entry on oeis.org

233, 89, 178, 34, 267, 123, 212, 68, 157, 13, 246, 102, 191, 47, 280, 136, 225, 81, 170, 26, 259, 115, 204, 60, 293, 149, 5, 238, 94, 183, 39, 272, 128, 217, 73, 162, 18, 251, 107, 196, 52, 285, 141, 230, 86, 175, 31, 264, 120, 209, 65, 298, 154, 10, 243, 99, 188
Offset: 0

Views

Author

Kenneth J Ramsey, Sep 05 2007

Keywords

Comments

The sequence can also be built up from left to right directly (without having to make insertions) as follows:
a(0) equals greatest odd Fibonacci number less than n, i.e., [a(0) = F(2m)]
The rule for a(n+1) is according to the following (first listed takes priority):
a(n+1) = a(n) + F(2m) if less than or equal to n
a(n+1) = a(n) - F(2m-1) if greater than 0
a(n+1) = a(n) + F(2m-2)
Continue until all n terms have been included in the sequence.

Examples

			4 appears between 2 and 1 in the sequence because the greatest odd Fibonacci number less than 4 is 2 and the greatest even Fibonacci number less than 4 is 3

Links

Crossrefs

Cf. A132828.

Formula

The sequence is generated starting with {2,1} and the numbers 3,4,5,..n are inserted in order into the sequence using the following rules: If n is an even Fibonacci number, it is inserted after the last term If n is an odd Fibonacci number, it is inserted before the first term If n is not a Fibonacci number, it is inserted between the adjacent terms, n - GF(even) and n-GF(odd) where GF(odd) and GF(even) are respectively the greatest odd and even Fibonacci numbers less than n.

A133117 Fractal sequence based on comparison of {n * tau} with {i*tau} for i = 1 to F(2j) where F(2j) equals the first i for which {n*tau} <= {i*tau} as i goes from 1 to F(2j+2)-1 and F(2j) equals the insertion point of n into P(n-1). The fractional parts {i*tau} are all less than or equal to {F(2j-2)*tau} for 0 < i < F(2j), so there is no chance that an insertion point greater than n in the permutation of the first n-1 integers will be specified by this rule. The table, A132827, gives the insertion points for each n into the permutation P(n-1) of the first n integers.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 5, 4, 2, 1, 3, 5, 4, 6, 2, 1, 3, 7, 5, 4, 6, 2, 1, 3, 7, 5, 4, 6, 2, 1, 3, 8
Offset: 1

Views

Author

Kenneth J Ramsey, Sep 13 2007

Keywords

Comments

This sequence is a modification of that in A054065 which gives the fractal series of the same permutation as the permutation of A132917 for which a couple of generating algorithms are given.

Examples

			The first few permutations are 1, 21, 213, 4213, 54213, 546213 since {6*tau} is greater than {1*Tau} but less than {3*Tau}; and since of 0<i<7 only {3*tau} and {6*tau} are greater than {1*tau}

Crossrefs

Cf. A054065, A132827, A132917, A132828.

Formula

See A132827.
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