A064885 Eisenstein array Ei(3,2).

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

Offset: 1

Author

Wolfdieter Lang, Oct 19 2001

Keywords

• nonn
• easy
• tabf

Comments

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

Examples

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

Links

• Index entries for sequences related to Stern's sequences

Crossrefs

Cf. A002487, A007051, A049456, A064881.

Programs

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] = 3; a[1, 1] = 2; Flatten[Table[a[n, m], {n, 1, nmax}, {m, 0, 2^(n - 1)}]] (* Jean-François Alcover, Sep 28 2011 *)

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) = 3, a(1, 1) = 2.