OFFSET
1,2
COMMENTS
In Eisenstein's notation this is the array for m=1 and n=2; see example in given reference p. 42. 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 3*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 and the link) with root 1/2. The composition rule for this tree is i/j -> i/(i+j), (i+j)/j.
LINKS
R. Backhouse, J. F. Ferreira, On Euclid’s algorithm and elementary number theory, Sci. Comput. Program. 76, No. 3, 160-180 (2011).
N. Calkin and H. S. Wilf, Recounting the Rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.
F. G. M. Eisenstein, Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen abhaengen und durch gewisse lineare Funktional-Gleichungen definirt werden, Verhandlungen der Koenigl. Preuss. Akademie der Wiss. Berlin (1850) 36-42, Feb 18, 1850. Werke, II, pp. 705-711.
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)=1, a(1, 1)=2.
EXAMPLE
{1,2};
{1,3,2};
{1,4,3,5,2};
{1,5,4,7,3,8,5,7,2}; ...
This binary subtree of rationals is built from
1/2;
1/3, 3/2;
1/4, 4/3, 3/5, 5/2; ...
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] = 1; a[1, 1] = 2; Flatten[ Table[a[n, m], {n, 1, nmax}, {m, 0, 2^(n - 1)}]] (* Jean-François Alcover, Sep 27 2011 *)
eisen = Most@Flatten@Transpose[{#, # + RotateLeft[#]}] &;
Flatten@NestList[eisen, {1, 2}, 6] (* Harlan J. Brothers, Feb 18 2015 *)
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Oct 19 2001
STATUS
approved