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

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

0,2

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

Links

Table of n, a(n) for n=0..35.

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.

Example

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

Mathematica

(See Conjecture under Comments.)

Crossrefs

Cf. A191426.

Sequence in context: A191440 A191727 A191537 * A126315 A125976 A071654

Adjacent sequences: A132824 A132825 A132826 * A132828 A132829 A132830

Keyword

nonn,tabl,uned

Author

Kenneth J Ramsey, Sep 03 2007

Status

approved