

A064883


Eisenstein array Ei(1,3).


1



1, 3, 1, 4, 3, 1, 5, 4, 7, 3, 1, 6, 5, 9, 4, 11, 7, 10, 3, 1, 7, 6, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 17, 10, 13, 3, 1, 8, 7, 13, 6, 17, 11, 16, 5, 19, 14, 23, 9, 22, 13, 17, 4, 19, 15, 26, 11, 29, 18, 25, 7, 24, 17, 27, 10
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

In Eisenstein's notation this is the array for m=1 and n=3; see pp. 412 of the Eisenstein reference given for A064881. The array for m=n=1 is A049456.
For n >= 1, the number of entries of row n is 2^(n1)+1 with the difference sequence [2,1,2,4,8,16,...]. Row sums give 4*A007051(n1).
The binary tree built from the rationals a(n,m)/a(n,m+1), m=0..2^(n1), for each row n >= 1 gives the subtree of the (Eisenstein)SternBrocot tree in the version of, e.g., Calkin and Wilf (for the reference see A002487, also for the Wilf link) with root 1/3. The composition rule for this tree is i/j > i/(i+j), (i+j)/j.


LINKS



FORMULA

a(n, m) = a(n1, m/2) if m is even, else a(n, m) = a(n1, (m1)/2) + a(n1, (m+1)/2), a(1, 0)=1, a(1, 1)=3.


EXAMPLE

{1,3}; {1,4,3}; {1,5,4,7,3}; {1,6,5,9,4,11,7,10,3}; ...
This binary subtree of rationals is built from 1/3; 1/4, 4/3; 1/5, 5/4, 4/7, 7/3; ...


MATHEMATICA

nmax = 6; a[n_, m_?EvenQ] := a[n1, m/2]; a[n_, m_?OddQ] := a[n, m] = a[n1, (m1)/2] + a[n1, (m+1)/2]; a[1, 0] = 1; a[1, 1] = 3; Flatten[ Table[ a[n, m], {n, 1, nmax}, {m, 0, 2^(n1)}]] (* JeanFrançois Alcover, Oct 03 2011 *)


CROSSREFS



KEYWORD

nonn,easy,tabf


AUTHOR



STATUS

approved



