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

%I A132827 #14 Jan 21 2020 00:11:31
%S A132827 1,3,2,8,6,4,21,16,11,5,55,42,29,14,7,144,110,76,37,19,9,377,288,199,
%T A132827 97,50,24,10,987,754,521,254,131,63,27,12
%N A132827 Table based upon insertion points of n into sequence A132828 and having a specific formula.
%C A132827 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.
%C A132827 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.
%C A132827 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.
%C A132827 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).
%C A132827 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
%F A132827 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.
%e A132827 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.
%e A132827 Corner of the array:
%e A132827   1,  3,  8, 21,  55
%e A132827   2,  6, 16, 42, 110
%e A132827   4, 11, 29, 76, 199
%e A132827   5, 14, 37, 97, 254
%t A132827 (See Conjecture under Comments.)
%Y A132827 Cf. A191426.
%K A132827 nonn,tabl,uned
%O A132827 0,2
%A A132827 _Kenneth J Ramsey_, Sep 03 2007