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A064881 Eisenstein array Ei(1,2). 9
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 (list; graph; refs; listen; history; text; internal format)
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 sub-tree 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.

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-Fran├žois 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 A300670 A137679

Adjacent sequences:  A064878 A064879 A064880 * A064882 A064883 A064884

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Oct 19 2001

STATUS

approved

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Last modified January 23 02:40 EST 2019. Contains 319365 sequences. (Running on oeis4.)