A064881 Eisenstein array Ei(1,2).

1, 2, 1, 3, 2, 1, 4, 3, 5, 2, 1, 5, 4, 7, 3, 8, 5, 7, 2, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 1, 7, 6, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 17, 10, 13, 3, 14, 11, 19, 8, 21, 13, 18, 5, 17, 12, 19, 7, 16, 9, 11, 2

Offset

1,2

Comments

In Eisenstein's notation this is the array for m=1 and n=2; see example in given reference p. 42. 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 3*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 and the link) with root 1/2. The composition rule for this tree is i/j -> i/(i+j), (i+j)/j.

Links

Table of n, a(n) for n=1..69.

R. Backhouse, J. F. Ferreira, On Euclid’s algorithm and elementary number theory, Sci. Comput. Program. 76, No. 3, 160-180 (2011).

N. Calkin and H. S. Wilf, Recounting the Rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.

F. G. M. Eisenstein, Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen abhaengen und durch gewisse lineare Funktional-Gleichungen definirt werden, Verhandlungen der Koenigl. Preuss. Akademie der Wiss. Berlin (1850) 36-42, Feb 18, 1850. Werke, II, pp. 705-711.

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

Example

{1,2};
{1,3,2};
{1,4,3,5,2};
{1,5,4,7,3,8,5,7,2}; ...
This binary subtree of rationals is built from
1/2;
1/3, 3/2;
1/4, 4/3, 3/5, 5/2; ...

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] = 1; a[1, 1] = 2; Flatten[ Table[a[n, m], {n, 1, nmax}, {m, 0, 2^(n - 1)}]] (* Jean-François Alcover, Sep 27 2011 *)
eisen = Most@Flatten@Transpose[{#, # + RotateLeft[#]}] &;
Flatten@NestList[eisen, {1, 2}, 6] (* Harlan J. Brothers, Feb 18 2015 *)

Crossrefs

Sequence in context: A272464 A133404 A134627 * A131967 A358120 A329501

Adjacent sequences: A064878 A064879 A064880 * A064882 A064883 A064884

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Oct 19 2001

Status

approved