A064882 Eisenstein array Ei(2,1).

2, 1, 2, 3, 1, 2, 5, 3, 4, 1, 2, 7, 5, 8, 3, 7, 4, 5, 1, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6, 1, 2, 11, 9, 16, 7, 19, 12, 17, 5, 18, 13, 21, 8, 19, 11, 14, 3, 13, 10, 17, 7, 18, 11, 15, 4, 13, 9, 14, 5, 11, 6, 7, 1

Offset

1,1

Comments

In Eisenstein's notation this is the array for m=2 and n=1; see pp. 41-2 of the Eisenstein reference given for A064881. This is identical with the array for m=1, n=2, given in A064881, 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 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 (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 2/1. The composition rule of this tree is i/j -> i/(i+j), (i+j)/j.

Links

Table of n, a(n) for n=1..69.

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)=2, a(1, 1)=1.

Example

{2,1}; {2,3,1}; {2,5,3,4,1}; {2,7,5,8,3,7,4,5,1}; ...
This binary subtree of rationals is built from 2/1; 2/3, 3/1; 2/5, 5/3, 3/4, 4/1; ...

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] = 2; a[1, 1] = 1; Flatten[ Table[ a[n, m], {n, 1, nmax}, {m, 0, 2^(n-1)}]] (* Jean-François Alcover, Sep 28 2011 *)

Crossrefs

Sequence in context: A131879 A172288 A134628 * A065158 A364842 A181842

Adjacent sequences: A064879 A064880 A064881 * A064883 A064884 A064885

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Oct 19 2001

Status

approved