A064884 Eisenstein array Ei(3,1).

3, 1, 3, 4, 1, 3, 7, 4, 5, 1, 3, 10, 7, 11, 4, 9, 5, 6, 1, 3, 13, 10, 17, 7, 18, 11, 15, 4, 13, 9, 14, 5, 11, 6, 7, 1, 3, 16, 13, 23, 10, 27, 17, 24, 7, 25, 18, 29, 11, 26, 15, 19, 4, 17, 13, 22, 9, 23, 14, 19, 5, 16, 11, 17, 6, 13

Offset

1,1

Comments

In Eisenstein's notation this is the array for m=3 and n=1; see pp. 41-42 of the Eisenstein reference given for A064881. This is identical with the array for m=1, n=3, given in A064883, 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 4*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 3/1. The composition rule of this tree is i/j -> i/(i+j), (i+j)/j.

Links

Table of n, a(n) for n=1..66.

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)=1.

Example

Array begins
  {3,  1};
  {3,  4,  1};
  {3,  7,  4,  5,  1};
  {3, 10,  7, 11,  4,  9,  5,  6,  1}; ...
This binary subtree of rationals is built from
  3/1;
  3/4, 4/1;
  3/7, 7/4, 4/5, 5/1; ...

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] = 3; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, nmax}, {m, 0, 2^(n-1)}]] (* Jean-François Alcover, Sep 28 2011 *)
eisen = Most@Flatten@Transpose[{#, # + RotateLeft[#]}] &;
Flatten@NestList[eisen, {3, 1}, 6] (* Harlan J. Brothers, Feb 18 2015 *)

Crossrefs

Sequence in context: A104765 A308690 A329512 * A093560 A173934 A131504

Adjacent sequences: A064881 A064882 A064883 * A064885 A064886 A064887

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Oct 19 2001

Status

approved