A064886 Eisenstein array Ei(2,3).

2, 3, 2, 5, 3, 2, 7, 5, 8, 3, 2, 9, 7, 12, 5, 13, 8, 11, 3, 2, 11, 9, 16, 7, 19, 12, 17, 5, 18, 13, 21, 8, 19, 11, 14, 3, 2, 13, 11, 20, 9, 25, 16, 23, 7, 26, 19, 31, 12, 29, 17, 22, 5, 23, 18, 31, 13, 34, 21, 29, 8, 27, 19, 30

Offset

1,1

Comments

In Eisenstein's notation this is the array for m=2 and n=3; see pp. 41-2 of the Eisenstein reference given for A064881. This is identical with the array for m=3, n=2, given in A064885, read backwards. The array for m=n=1 is A049456.

For n >= 1, the number of entries of row n >= 1 is 2^(n-1)+1 with the difference sequence [2,1,2,4,8,16,...]. Row sums give 5*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 2/3. The composition rule of this tree is i/j -> i/(i+j), (i+j)/j.

Links

Table of n, a(n) for n=1..64.

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)=3.

Example

{2,3}; {2,5,3}; {2,7,5,8,3}; {2,9,7,12,5,13,8,11,3}; ...
This binary subtree of rationals is built from 2/3; 2/5,5/3; 2/7,7/5,5/8,8/3; ...

Mathematica

a[1, 0] = 2; a[1, 1] = 3; a[n_ /; n >= 1, m_ /; m >= 0] := If[EvenQ[m], a[n, m] = a[n-1, m/2], a[n, m] = a[n-1, (m-1)/2] + a[n-1, (m+1)/2]]; Table[a[n, m], {n, 1, 6}, {m, 0, 2^(n-1)}] // Flatten (* Jean-François Alcover, Feb 27 2018 *)

Crossrefs

Sequence in context: A302170 A049805 A104887 * A029600 A169616 A344448

Adjacent sequences: A064883 A064884 A064885 * A064887 A064888 A064889

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Oct 19 2001

Status

approved