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

1,1

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.

Links

Table of n, a(n) for n=1..63.

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

Example

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; ...

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 *)

Crossrefs

Cf. A002487, A007051, A049456, A064881.

Sequence in context: A173093 A236361 A227634 * A029618 A330656 A264399

Adjacent sequences: A064882 A064883 A064884 * A064886 A064887 A064888

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Oct 19 2001

Status

approved