0,2

The numbers n in column j of this table always have (F(2j) -1) numbers less than n that appear befor 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 an 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 how many terms to the left of each number in the array 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)

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.

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.

Sequence in context: A125980 A126320 A135992 this_sequence A126315 A125976 A071654

Adjacent sequences: A132824 A132825 A132826 this_sequence A132828 A132829 A132830

nonn,tabl,uned

Kenneth J Ramsey (Ramsey2879(AT)msn.com), Sep 03 2007