A064883 Eisenstein array Ei(1,3).

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

Offset

1,2

Comments

In Eisenstein's notation this is the array for m=1 and n=3; see pp. 41-2 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^(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 1/3. The composition rule for this tree is i/j -> i/(i+j), (i+j)/j.

Links

Table of n, a(n) for n=1..65.

Index entries for sequences related to Stern's sequences

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)=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[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] = 3; Flatten[ Table[ a[n, m], {n, 1, nmax}, {m, 0, 2^(n-1)}]] (* Jean-François Alcover, Oct 03 2011 *)

Crossrefs

Sequence in context: A055171 A337209 A101038 * A090844 A008314 A104568

Adjacent sequences: A064880 A064881 A064882 * A064884 A064885 A064886

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Oct 19 2001

Status

approved