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

Author

Wolfdieter Lang, Oct 19 2001

Keywords

• nonn
• easy
• tabf

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.

Examples

			{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; ...
		

Links

• Index entries for sequences related to Stern's sequences

Programs

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 *)

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.