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
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
KEYWORD
AUTHOR
Kenneth J Ramsey, Sep 03 2007
STATUS
approved