OFFSET
1,1
COMMENTS
In Eisenstein's notation this is the array for m=3 and n=1; see pp. 41-42 of the Eisenstein reference given for A064881. This is identical with the array for m=1, n=3, given in A064883, read backwards. The array for m=n=1 is A049456.
For n >= 1, the number of entries of row n is 2^(n-1)+1 with the difference sequence [2,1,2,4,8,16,...]. Row sums give 4*A007051(n-1).
The binary tree built from the rationals a(n,m)/a(n,m+1), m=0..2^(n-1), for each row n >= 1 gives the subtree of the (Eisenstein-)Stern-Brocot tree in the version of, e.g., Calkin and Wilf (for the reference see A002487, also for the Wilf link) with root 3/1. The composition rule of this tree is i/j -> i/(i+j), (i+j)/j.
FORMULA
a(n, m) = a(n-1, m/2) if m is even, else a(n, m) = a(n-1, (m-1)/2) + a(n-1, (m+1)/2), a(1, 0)=3, a(1, 1)=1.
EXAMPLE
Array begins
{3, 1};
{3, 4, 1};
{3, 7, 4, 5, 1};
{3, 10, 7, 11, 4, 9, 5, 6, 1}; ...
This binary subtree of rationals is built from
3/1;
3/4, 4/1;
3/7, 7/4, 4/5, 5/1; ...
MATHEMATICA
nmax = 6; a[n_, m_?EvenQ] := a[n-1, m/2]; a[n_, m_?OddQ] := a[n, m] = a[n-1, (m-1)/2] + a[n-1, (m+1)/2]; a[1, 0] = 3; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, nmax}, {m, 0, 2^(n-1)}]] (* Jean-François Alcover, Sep 28 2011 *)
eisen = Most@Flatten@Transpose[{#, # + RotateLeft[#]}] &;
Flatten@NestList[eisen, {3, 1}, 6] (* Harlan J. Brothers, Feb 18 2015 *)
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Oct 19 2001
STATUS
approved