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\begin{center}
\vskip 1cm{\LARGE\bf
A Multidimensional Continued Fraction \\
\vskip .1in
Generalization of Stern's Diatomic Sequence
}
\vskip 1cm
\large
Thomas Garrity\\
Department of Mathematics and Statistics\\
Williams College\\
Williamstown, MA 01267\\
USA \\
\href{mailto:tgarrity@williams.edu}{\tt tgarrity@williams.edu}
\end{center}
\vskip .2 in
\begin{abstract}
Continued fractions are linked to Stern's diatomic sequence
$0,1,1,2,1,3,2,3,1,4,\ldots $ (given by the recursion relations
$\alpha_{2n}=\alpha_n$ and $\alpha_{2n+1} = \alpha_n + \alpha_{n+1},$
where $\alpha_0=0$ and $\alpha_1=1$), which has long been known. Using
a particular multidimensional continued fraction algorithm (the Farey
algorithm), we generalize the diatomic sequence to a sequence of
numbers that quite naturally can be termed Stern's triatomic
sequence (or a two-dimensional Pascal sequence with memory). As
both continued fractions and the diatomic sequence can be thought of as
coming from a systematic subdivision of the unit interval, this new
triatomic sequence arises by a systematic subdivision of a
triangle. We discuss some of the algebraic properties of the
triatomic sequence.
\end{abstract}
\def\R{{\mathbb R}} % Reals
\def\C{{\mathbb C}} % Complex
\def\P{{\mathbb P}} % Projective Space
\def\A{{\mathbb A}} % Affine Space
\def\Z{{\mathbb Z}} % Integers
\def\N{{\mathbb N}} % natural numbers
\section{The classical Stern diatomic sequence}
The goal of this paper is to take a particular generalization of continued fractions (the Farey algorithm) and find the corresponding generalization of Stern's diatomic sequence, which we will call Stern's triatomic sequence.
In this first section, though, we will
quickly review the basics of Stern's diatomic sequence (sequence
\seqnum{A002487} in Sloane's {\it Encyclopedia}). Hence experts should skip to the next section.
In Section~\ref{sec:Stern's Triatomic sequence} we give a formal presentation of Stern's triatomic sequence. Section~\ref{sec:Stern's Triatomic Sequence and Triangles} shows how Stern's triatomic sequence can be derived from a systematic subdivision of a triangle. Section~\ref{sec:The Farey Map}
shows how Stern's triatomic sequence arises from the multidimensional continued fraction stemming from the Farey map, in analog to the link between Stern's diatomic sequence and classical continued fractions. Sections~\ref{sec:Combinatorial Properties of Tri-Atomic Sequences} and \ref{sec:Paths of a directed graph} discuss some of the properties for this sequence. Many of these properties are analogs of corresponding properties for Stern's diatomic sequence. We will conclude with open questions.
A recent {\it Monthly\/} article by Northshield \cite{Northshield1} gives a good overview of Stern's diatomic sequence and, more importantly for this paper, its relationship to continued fractions. In much earlier work,
Lehmer \cite{Lehmer1} lists a number of properties of this sequence. In part, our eventual goal is showing how these properties usually generalize for our new Stern's triatomic sequences.
Consider an interval, with endpoints weighted by numbers $v_1$ and $v_2$:
\begin{center}
\begin{tikzpicture}
\draw[|-|] (0,0)--(5,0);
\node[below] at (0,0){$v_1$};
\node[below] at (5,0){$v_2$};
\end{tikzpicture}
\end{center}
Then subdivide:
\begin{center}
\begin{tikzpicture}
\draw[|-|] (0,0)--(2.5,0);
\draw[|-|] (5,0)--(2.5,0);
\node[below] at (0,0){$v_1$};
\node[below] at (5,0){$v_2$};
\node[below] at (2.5,0){$v_1 + v_2$};
\end{tikzpicture}
\end{center}
We can continue, getting:
\begin{center}
\begin{tikzpicture}
\draw[|-|] (0,0)--(1.66,0);
\draw[|-|] (1.66,0)--(2.5,0);
\draw[|-|] (3.38,0)--(2.5,0);
\draw[|-|] (3.38,0)--(5,0);
\node[below] at (0,0){$v_1$};
\node[below] at (5,0){$v_2$};
\node[below] at (2.5,0){$v_1 + v_2$};
\node[above] at (1.66,0){$2v_1+v_2$};
\node[above] at (3.38,0){$v_1+2v_2$};
\end{tikzpicture}
\end{center}
The traditional Stern diatomic sequence is doing this for the case when $v_1=v_2=1$. If instead we start with vectors
$$v_1= \left( \begin{array}{c} 0 \\ 1 \end{array} \right) ,\; v_2=\left( \begin{array}{c} 1 \\ 1 \end{array} \right) $$
and do the standard identification of
$$\left( \begin{array}{c} x \\ y \end{array} \right) \rightarrow \frac{x}{y},$$
we get the Farey decomposition of the unit interval, which in turn can be used to recover the continued fraction expansion of any real number on the unit interval. The case of letting $v_1=v_2=1$ is equivalent to keeping track of the denominators for the Farey decomposition. (We mention this as motivation for why we later link our new Stern's triatomic sequence to a particular multidimensional continued fraction algorithm.)
Thus Stern's diatomic sequence is given by the recursion formulas
$$\alpha_0= 0, \alpha_1=1$$
and
\begin{eqnarray*}
\alpha_{2n} &=& \alpha_n \\
\alpha_{2n+1} &=& \alpha_n + \alpha_{n+1}
\end{eqnarray*}
The first few terms are
$$0,1,1,2,1,3,2,3,1,4,3,5,2,5,4,\ldots.$$
It is standard to write this as
$$ \begin{array} {ccccccccc}
\alpha_1 & & & & & & & & \alpha_2 \\
\alpha_2 & & & & \alpha_3 & & & & \alpha_4 \\
\alpha_4 & & \alpha_5 & & \alpha_6 & & \alpha_7 & & \alpha_8 \\
\alpha_8 & \alpha_9 & \alpha_{10} & \alpha_{11} & \alpha_{12} & \alpha_{13} & \alpha_{14} & \alpha_{15} & \alpha_{16} \\
\end{array}$$
For $\alpha_1=1$, we get
$$ \begin{array} {ccccccccccccccccc}
1 && && & & && && && && && 1 \\
1 && && && && 2 && && && && 1 \\
1 && && 3 && && 2 && && 3 && && 1 \\
1 && 4 && 3 && 5 && 2 && 5 && 3 && 4 && 1 \\
1 &5& 4 &7& 3 &8& 5 &7& 2 &7& 5 &8& 3 &7& 4 &5& 1 \\
\end{array}$$
Thus to create the next level, we simply interlace the original level with the sum of adjacent entries. This is why this sequence is called by
Knauf ``Pascal with memory'' by Knauf \cite{Knauf1}. (It was Knauf's work on linking this sequence with statistical mechanics, which is also in \cite{Kelban-Ozluk1, Knauf5, Knauf2, Knauf3, Knauf4}, that led the author to look at these sequences. Other work linking statistical mechanics with Stern diatomic sequences, though these works do not use the rhetoric of Stern diatomic sequences, include \cite{Contucci-Knauf1, Esposti-Isola-Knauf1, Fiala-Kleban1, Fiala-Kleban-Ozluk1, Garrity10, Guerra-Knauf1, Kallies-Ozluk-Peter-Syder1, Kelban-Ozluk1, Mayer2, MendesFrance-Tenenbaum1, MendesFrance-Tenenbaum2, Prellberg1, Prellberg-Fiala-Kleban1, Prellberg-Slawny1}).
A matrix approach for Stern's diatomic sequence $\alpha_1, \alpha_2,
\ldots $ is as follows. Set
$$b_0= \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right), b_1= \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right)$$
and
$$M= \left( \begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array} \right).$$
Let $N >2$ be a positive integer. Then there is a tuple $(i_k, \ldots, i_1)_S$ with $k$ a positive integer and each $i_j\in \{0,1\}$ such that
$$N=2^{k} + 1 + i_{k}2^{k-1}+\cdots + i_1.$$
(The subscript $S$ stands for `Stern'.) For example $3 = 2^1 + 1 + 0\cdot 1 = (0)_S,
4 = 2+ 1 + 1\cdot 1 = (1)_S,
5 = 2^2 + 1+ + 0 \cdot 2^1 + 0\cdot 1 = (0,0)_S $, $6=2^2 +1+ 0\cdot 2 + 1\cdot 1 = (0,1)_S$, $7= 2^2 + 1+ 1\cdot 2 + 0 \cdot 1 = (1,0)_S$, $8=(1,1)_S$ and
$16 = 2^3 + 1 + 1 \cdot 2^2 + 1\cdot 2^1 + 1\cdot 1= (1,1, 1)_S$.
Then we have $$\alpha_N = \left( \begin{array}{cc} 0 & 1 \end{array} \right) Mb_{i_k}\cdots b_{i_1} \left( \begin{array}{c} 0 \\ 1 \end{array} \right) .$$
This is just systemizing the subdivision of the unit interval via Farey fractions. As described by the author \cite{Garrity10}, these matrices can be used to find the continued fraction expansion for any real number, which is in part why one would expect any natural generalization of Stern's diatomic sequence to be linked with a generalization of continued fractions.
Finally, Allouche and Shallit \cite{Allouche-Shallit92, Allouche-Shallit03} have shown that Stern's diatomic sequence is a $2$-regular sequence.
\section{Stern's triatomic sequence} \label{sec:Stern's Triatomic sequence}
We now start with explaining the new sequence of this paper.
Start with an initial triple (which we will also think of as a $1\times 3$ row matrix)
$$\triangle = (v_1, v_2, v_3) = (1,1,1).$$
We are using the notation $\triangle$ to suggest the vertices of a triangle. For any index
$$I=(i_1, i_2, \ldots, i_n),$$
with each $i_k$ being zero, one or two, we will associate a triple which we denote by $\triangle(I)$.
Suppose we know the triple
$$\triangle(I)=(v_1(I),v_2(I),v_3(I)).$$
Consider the three matrices:
$$A_0= \left( \begin{array}{rrr} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array} \right), A_1= \left( \begin{array}{rrr} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{array} \right) A_2= \left( \begin{array}{rrr} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 1 & 0 & 1 \end{array} \right).$$
Then we set
$$\begin{array}{ccccc}
\triangle (I,0) &=& \triangle (I) A_0 &=&(v_1(I),v_2(I),v_3(I)) A_0 \\
\triangle (I,1) &=& \triangle (I) A_1 &=&(v_1(I),v_2(I),v_3(I)) A_1 \\
\triangle (I,2) &=& \triangle (I) A_2 &=&(v_1(I),v_2(I),v_3(I)) A_2.
\end{array}$$
Thus we have
$$\begin{array}{rclrclrcl}
v_1(I,0) &=& v_1(I), & v_2(I,0) &=& v_2(I), &
v_3(I,0) &=& v_1(I) + v_2(I) + v_3(I) \\
v_1(I,1) &=& v_2(I), &
v_2(I,1) &=& v_3(I), &
v_3(I,1) &=& v_1(I) + v_2(I) + v_3(I) \\
v_1(I,2) &=& v_3(I), &
v_2(I,2) &=& v_1(I), &
v_3(I,2) &=& v_1(I) + v_2(I) + v_3(I).
\end{array}$$
Thus for any $I=(i_1,i_2, \ldots, i_n)$, we have
\begin{eqnarray*}
\triangle(I) &=& (v_1(I),v_2(I),v_3(I)) \\
&=& (1,1,1)A_{i_1}\cdots A_{i_n} \\
&=& (v_1,v_2,v_3))A_{i_1}\cdots A_{i_n} \\
&=& \triangle A_{i_1}\cdots A_{i_n}
\end{eqnarray*}
For $I=(i_1, \ldots i_n)$ and $J=(j_1, \ldots , j_m)$, we say that $I, dashed]( 5,2)--(2.6,2);
\node[below left] at (0,0){$v_1$};
\node[below right] at (5,0){$v_2$};
\node[above] at (2.5,4){$v_3$};
\node[right] at (5, 2){$v_1+v_2+v_3$};
\end{tikzpicture}
\end{center}
\noindent where $\triangle(0)$ has vertices $v_1, v_2, v_1+v_2+v_3$, $\triangle(1)$ has vertices $v_2, v_3, v_1+v_2+v_3$ and $\triangle(2)$ has vertices $v_3, v_1, v_1+v_2+v_3$.
Each of these triangles can be divided into three more triangles, giving us nine triangles
\begin{center}
\begin{tikzpicture}
\draw(2.5,4)--(0,0);
\draw(2.5,4)--(5,0);
\draw (0,0)--(5,0);
\draw(2.5,1.8)--(2.5,4);
\draw(2.5,1.8)--(0,0);
\draw(2.5,1.8)--(5,0);
\draw[ ultra thick](2.5, .9)--(2.5,1.8);
\draw[ ultra thick] (2.5, .9)--(0,0);
\draw[ ultra thick](2.5, .9)--(5,0);
\draw[ ultra thick](1.8, 1.9)--(2.5, 1.8);
\draw[ ultra thick](1.8, 1.9)--(0,0);
\draw[ ultra thick](1.8, 1.9)--(2.5, 4);
\draw[ ultra thick](3.2, 1.9)--(5,0);
\draw[ ultra thick](3.2, 1.9)--(2.5,4);
\draw[ ultra thick](3.2, 1.9)--(2.5,1.8);
\draw[->, dashed]( 5,1.6)--(2.7,1.75);
\draw[->, dashed]( 4.2,3.1)--(3.3, 2);
\draw[->, dashed](.8, 3.1)--(1.7, 2);
\draw[->, dashed](2.5, -.5)--(2.5, .8);
\node[below left] at (0,0){$v_1$};
\node[below right] at (5,0){$v_2$};
\node[above] at (2.5,4){$v_3$};
\node[right] at (5, 1.6){$v_1+v_2+v_3$};
\node[above right] at (4.2, 3.1){$v_1+2v_2+2v_3$};
\node[above left] at (.8, 3.1){$2v_1+v_2+2v_3$};
\node[below] at (2.5, -.5){$2v_1+v_2+2v_3$};
\end{tikzpicture}
\end{center}
We will identify each of these triangles with their vertices. Thus we start with our initial triangle (which we will say is at level $0$):
$$\triangle =(v_1, v_2, v_3).$$
At level one we have
\begin{eqnarray*}
\triangle(0) & = & (v_1, v_2, v_1+v_2+ v_3) \\
\triangle(1) & = & (v_2, v_3, v_1+v_2+ v_3) \\
\triangle(2) & = & (v_3, v_1, v_1+v_2+ v_3)
\end{eqnarray*}
\begin{center}
\begin{tikzpicture}
\draw(2.5,1.8)--(2.5,4);
\draw(2.5,4)--(0,0);
\draw(2.5,4)--(5,0);
\draw (0,0)--(5,0);
\draw(2.5,1.8)--(0,0);
\draw(2.5,1.8)--(5,0);
\node at (2.5, .9){$\triangle (0)$};
\node at (3.2,1 .9){$\triangle (1)$};
\node at (1.8,1 .9){$\triangle (2)$};
\node[below left] at (0,0){$v_1$};
\node[below right] at (5,0){$v_2$};
\node[above] at (2.5,4){$v_3$};
\end{tikzpicture}
\end{center}
while for level two we have
\begin{eqnarray*}
\triangle(00) & = & (v_1, v_2, 2v_1+ 2 v_2+ v_3) \\
\triangle(01) & = & (v_2, v_1 + v_2 + v_3, 2v_1+ 2 v_2+ v_3) \\
\triangle(02) & = & (v_1+v_2+v_3, v_1, 2v_1+ 2 v_2+ v_3) \\
\triangle(10) & = & (v_2, v_3, v_1+ 2 v_2+ 2v_3) \\
\triangle(11) & = & (v_3, v_1+v_2+ v_3, v_1+ 2 v_2+ 2v_3) \\
\triangle(12) & = & (v_1+v_2+ v_3, v_2, v_1+ 2 v_2+ 2v_3) \\
\triangle(20) & = & (v_3, v_1, 2v_1+ v_2+ 2v_3) \\
\triangle(21) & = & (v_1, v_1+v_2+ v_3, 2v_1+ v_2+ 2v_3) \\
\triangle(21) & = & ( v_1+v_2+ v_3, v_3, 2v_1+ v_2+ 2v_3).
\end{eqnarray*}
\begin{center}
\begin{tikzpicture}[scale=1.4]
\draw(2.5,4)--(0,0);
\draw(2.5,4)--(5,0);
\draw (0,0)--(5,0);
\draw(2.5,1.8)--(2.5,4);
\draw(2.5,1.8)--(0,0);
\draw(2.5,1.8)--(5,0);
\draw(2.5, .9)--(2.5,1.8);
\draw(2.5, .9)--(0,0);
\draw(2.5, .9)--(5,0);
\draw(1.8, 1.9)--(2.5, 1.8);
\draw(1.8, 1.9)--(0,0);
\draw(1.8, 1.9)--(2.5, 4);
\draw(3.2, 1.9)--(5,0);
\draw(3.2, 1.9)--(2.5,4);
\draw(3.2, 1.9)--(2.5,1.8);
\node at (2.5,.4){$\triangle(00)$};
\node at (3.1,1){$\triangle(01)$};
\node at (1.9,1){$\triangle(02)$};
\node[right] at (5,1.2){$\triangle(12)$};
\draw[->, dashed]( 5,1.2)--(3.1,1.6);
\node[left] at (0,1.2){$\triangle(21)$};
\draw[->, dashed]( 0,1.2)--(1.9,1.6);
\node[right] at (4.5,2.1){$\triangle(10)$};
\draw[->, dashed]( 4.5,2.1)--(3.5, 2);
\node[left] at (0.5,2.1){$\triangle(20)$};
\draw[->, dashed]( 0.5,2.1)--(1.5, 2);
\node[right] at (3.8, 3){$\triangle(11)$};
\draw[->, dashed]( 3.8,3)--(2.8, 2.4);
\node[left] at (1.2,3){$\triangle(22)$};
\draw[->, dashed]( 1.2,3)--(2.2, 2.4);
\node[below left] at (0,0){$v_1$};
\node[below right] at (5,0){$v_2$};
\node[above] at (2.5,4){$v_3$};
\end{tikzpicture}
\end{center}
Continuing this process produces a sequence of triples
$$\triangle, \triangle(0),\triangle(1), \triangle(2),\triangle(00),\triangle(01),\triangle(02), \triangle(10), \ldots, $$
which, in a natural way, will give us last section's Stern's triatomic sequence.
\section{The Farey map}\label{sec:The Farey Map}
\subsection{Definition}
Stern's diatomic sequence is linked to continued fractions \cite{Northshield1}. (This can also be seen in how the diatomic sequence can be interpreted via the Farey decomposition of the unit interval.) There is a multidimensional continued fraction algorithm which generates in an analogous fashion Stern's triatomic sequence.
Beaver and the author \cite{Beaver-Garrity1} gave a (seemingly) new multidimensional continued fraction algorithm, with the goal of finding finding a generalization of the Minkowski ?($x$) function. This algorithm, though, seems particularly well-suited to generalize Stern's diatomic sequence.
(Panti \cite{Panti1}
used the Monkmeyer algorithm to generalize the Minkowski ?$(x)$ function.) Also, it is standard in the study of multidimensional continued fractions to pass back and forth between cones in $\R^3$ and triangle in $\R^2$, via the map, when $x\neq 0$,
$(x,y,z) \rightarrow (y/x, z/x)$. In this subsection we define the Farey map as an iteration of a triangle, while in the next subsection we will think of the Farey map as an iteration of the corresponding cone.
Let $$\triangle = \{(x,y) \in \R^2: 0\leq y \leq x \leq 1\}.$$
Partition $\triangle $ into the three subtriangles
\begin{eqnarray*}
\triangle_0 & = & \{(x,y) \in \triangle : 1-2y \geq x-y \geq y \} \\
\triangle_1 & = & \{(x,y) \in \triangle : 2x-1 \geq y \geq 1-x \} \\
\triangle_2 & = & \{(x,y) \in \triangle : 1-2x+2y \geq 1-x \geq x-y \}.
\end{eqnarray*}
The Farey map $T:\triangle \rightarrow \triangle$ is given by three one-to-one onto maps
$$T_i:\triangle_i \rightarrow \triangle$$
defined via
\begin{eqnarray*}
T_0(x,y) & = & \left( \frac{x-y}{1-2y}, \frac{y}{1-2y} \right) \\
T_1(x,y) &=& \left( \frac{y}{2x-1}, \frac{1-x}{2x-1} \right) \\
T_2(x,y) &=& \left( \frac{1-x}{1-2x+2y}, \frac{x-y}{1-2x+2y} \right)
\end{eqnarray*}
These maps are the Farey analog of the Gauss map for continued fractions.
\subsection{Farey map on vertices}
Now we will identify triangles with their corresponding cones in $\R^3$. Let $\triangle$ be a cone with three vertices $v_1, v_2, v_3$, each written as a column vector in $\R^3$. Subdivide $\triangle$ into three sub-cones
$$\begin{array}{ccc}
\triangle(0) & \mbox{with vertices}& v_1, v_2,v_1+v_2+ v_3 \\
\triangle(1) & \mbox{with vertices}& v_2, v_3,v_1+v_2+ v_3 \\
\triangle(3) & \mbox{with vertices} &v_3, v_1, v_1+v_2+ v_3
\end{array}$$
We have three maps
$$A_i:\triangle \rightarrow \triangle_i,$$
which we describe via matrix multiplication on the right.
Recall that
$$A_0= \left( \begin{array}{ccc} 1 & 0 & 1\\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}
\right), A_1 = \left( \begin{array}{ccc} 0 & 0 & 1\\ 1 & 0 & 1 \\ 0 & 1 & 1\end{array}
\right), A_ 2 = \left( \begin{array}{ccc} 0 & 1 & 1\\ 0 & 0 & 1 \\ 1 & 0 & 1\end{array}
\right).$$
For any three vectors $v_1$ $v_2$ and $v_3$, we have
$$(v_1,v_2.v_3)A_0= (v_1, v_2, v_1 + v_2 + v_3)$$
$$(v_1,v_2.v_3)A_1= (v_2, v_3, v_1 + v_2 + v_3)$$
$$(v_1,v_2.v_3)A_2= (v_3, v_1, v_1 + v_2 + v_3).$$
We now specify our initial three vectors, by setting, as before,
$$V= (v_1,v_2,v_3)= \left(\begin{array}{ccc} 0 & 1 &1 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \end{array} \right).$$
Then $VA_0, VA_1$ and $VA_2$
is simply the Farey map's analog of the Farey subdivision of the unit interval and can be interpreted as the matrix approach for the inverse maps of the three Farey maps $T_i:\triangle_i\rightarrow \triangle$ of triangles in $\R^2$ from the previous subsection.
Then we have
\begin{theorem} For an $n$-tuple $I=(i_1,\ldots, i_n)$ where each $i_j$ is a zero, one or two, we have that
$$VA_{i_1}\ldots A_{i_n} = \left( \begin{array}{ccc} * & * & * \\ * & * & * \\ v_1(I)&v_ 2(I) & v_3(I) \end{array} \right).$$
\end{theorem}
Thus Stern's triatomic sequence is the analog to the fact that Stern's diatomic sequence can be interpreted as keeping track of the denominators for the Farey decomposition of the unit interval.
\section{Combinatorial properties of triatomic sequences}
\label{sec:Combinatorial Properties of Tri-Atomic Sequences}
We now develop some analogs of the combinatorial properties for Stern diatomic sequences for triatomic sequences.
\subsection{Three-fold symmetry}
Property 4 in Lehmer \cite{Lehmer1} is that there is a two-fold symmetry for Stern's diatomic sequences on each ``level''. We will see that for triatomic sequences, we have a three-fold symmetry at each level.
\begin{proposition} For any
$N=\tau(0, j_{2}, \ldots , j_{n};k),$ where each $j_i=0, 1$ or $2$ and $k$ is as large as possible for $k\in \{1,2,3\},$ we have
$$a_N = a_{N+ 3^k} = a_{N+ 2\cdot 3^N}.$$
\end{proposition}
Thus
$$a_{\tau(0, j_{2}, \ldots , j_{n};k)} = a_{\tau(1, j_{2}, \ldots , j_{n};k)} = a_{\tau(2, j_{2}, \ldots , j_{n},;k)} .$$
\begin{proof}
At the $0$th level, we have
$$a_1= 1, \;a_2=1, \; a_3=1.$$
The $1$th level is
$$a_4, \ldots , a_{12},$$
which by definition is
$$a_1, a_2, a_1 + a_2 + a_3, a_2, a_3, a_1 + a_2 + a_3, a_3, a_1, a_1 + a_2 + a_3,$$
which in turn is simply
$$1,1,3,1,1,3,1,1,3,$$
satisfying the desired relation.
The rest follows from the basic recursion relation.
\end{proof}
\subsection{Tribonacci numbers}
Recall that the Tribonacci numbers are the terms in the sequence $\beta_1, \beta_2, \ldots$ given by the recursion relation
$$\beta_{k+3} = \beta_k + \beta_{k+1} + \beta_{k+2}.$$
Such sequences are intimately intwined with triatomic sequences.
\begin{theorem} For any $n$-tuple $J$, the following subsequence is a Tribonacci sequence:
$$a_{\tau(J:1)}, a_{\tau(J:2)}, a_{\tau(J:3)}, a_{\tau(J, 1:3)}, a_{\tau(J, 1,1:3)}, a_{\tau(J, 1,1,1:3)}, \ldots$$
\end{theorem}
\begin{proof}
Start with the triple
$$a_{\tau(J:1)}, a_{\tau(J:2)}, a_{\tau(J:3)}.$$
Then the triple
$a_{\tau(J, 1:1)}, a_{\tau(J,1:2)}, a_{\tau(J,1:3)}$
is
$$a_{\tau(J:2)}, a_{\tau(J:3)}, a_{\tau(J:1)}+ a_{\tau(J:2)}+ a_{\tau(J:3)}.$$
Next, the triple $a_{\tau(J, 1,1:1)}, a_{\tau(J,1,1:2)}, a_{\tau(J,1,1:3)}$ is $$ a_{\tau(J:3)},a_\tau{(J,1:3)}, a_{\tau(J:2)}+ a_{(J:3)} + a_{(J,1:3)}.$$
This process continues, since by definition we have
$$a_{(J,1^s:3)} = a_{(J,1^{s-3}:3)} + a_{(J,1^{s-2}:3)} + a_{(J,1^{s-1}:3)},$$
where $1^s$ is notation for $s$ ones in a row,
giving us our result.
\end{proof}
\begin{corollary} The standard Tribonacci sequence $$1,1,1,3,5,9,17,31,57,105, 193,\ldots $$
is $$a_1, a_2, a_3, a_{\tau(1:3)}, a_{\tau(1,1:3)}, \ldots , a_{\tau(1^s:3)}, \ldots $$
\end{corollary}
\subsection{Sums at levels}
For Stern's diatomic sequence, Stern showed that the sum of all terms of level $n$ is $3^n+1$ (see property 2 in \cite{Lehmer1}). A similar sum exists for Stern's triatomic sequence.
\begin{proposition}The sum of all terms in Stern's triatomic sequence at the $n$th level is $5^n\cdot 3$. Thus for fixed $n$
$$\sum a_{\tau(J:k)} = 5^n \cdot 3,$$
where the summation is over all $n$-tuples $J= (j_1, \ldots j_n)$, for $j_i=0, 1$ or $2$, and over $k=1,2,3$.
\end{proposition}
\begin{proof} The $0$th level is the three numbers $1,1,1$, and hence the sum is $3$. Now suppose we have
at the $n$ level the number
$N=\tau(j_1, \ldots, j_n: 1)$.
Then the triple of numbers
$$a_N, a_{N+1}, a_{N+2}$$
is in the $n$th level. What is important is that this triple generates the following nine terms in the $(n+1)$st level:
$$a_N, a_{N+1}, a_N+ a_{N+1} + a_{N+2},$$
$$a_{N+1}, a_{N+2}, a_N+ a_{N+1} + a_{N+2}$$
and
$$a_{N+2}, a_{N}, a_N+ a_{N+1} + a_{N+2},$$
whose sum is
$$5(a_N+ a_{N+1} + a_{N+2}),$$
giving us our result.
\end{proof}
\subsection{Values at levels}
We can write each positive integer $N$ as $N=\tau(j_1, \ldots , j_n;k)$, where each $j_i\in \{0,1,2\}$ and each $k\in \{1,2,3\}$. We have determined what the corresponding $a_N$ is at the $n$th level. We set
$$\delta_n(m)= \mbox{number of times $m$ occurs in the $n$th level}.$$
We further set
$$\delta_n^1(m)= \mbox{number of times $m$ occurs in the $n$th level when $k=1$},$$
$$\delta_n^2(m)= \mbox{number of times $m$ occurs in the $n$th level when $k=2$},$$
and
$$\delta_n^3(m)= \mbox{number of times $m$ occurs in the $n$th level when $k=3$}.$$
At level $1$, our sequence is
$$1,1,3,1,1,3,1,1,3,$$
in which case
$$\delta_1(1) = 6, \delta_1(3) = 3 , \delta_1(n)=0\; \mbox{for} \; n\neq1,3,$$
$$\delta_1^1(1) = 3, \delta_1^1(3) = 1 , \delta_1^1(n)=0\; \mbox{for} \; n\neq1,3,$$
$$\delta_1^2(1) = 3, \delta_1^2(3) = 1 , \delta_1^2(n)=0\; \mbox{for} \; n\neq1,3,$$
$$\delta_1^3(1) = 0, \delta_1^2(3) = 3 , \delta_1^2(n)=0\; \mbox{for} \; n\neq1,3.$$
We now discuss some formulas for these new functions.
First for a technical lemma:
\begin{lemma}Suppose we are at the $n$th level, for some
$N=\tau(j_1, \ldots , j_n;1)$. Then the triple $$a_N,a_{N+1},a_{N+2}$$ will induce the following three triples on the $(n+1)$st level:
The triple corresponding to
$$a_{\tau(j_1, \ldots , j_n,0;1)}, a_{\tau(j_1, \ldots , j_n,0;2)}, a_{\tau(j_1, \ldots , j_n,0;3)}$$ will be
$$ (a_N, a_{N+1}, a_N+a_{N+1}+a_{N+2}), $$
the triple corresponding to
$$a_{\tau(j_1, \ldots , j_n,1;1)}, a_{\tau(j_1, \ldots , j_n,1;2)}, a_{(j_1, \ldots , j_n,1;3)}$$ will be
$$ (a_{N+1}, a_{N+2}, a_N+a_{N+1}+a_{N+2}), $$
and the triple corresponding to
$$a_{\tau(j_1, \ldots , j_n,2;1)}, a_{\tau(j_1, \ldots , j_n,2;2)}, a_{\tau(j_1, \ldots , j_n,2;3)}$$ will be
$$ (a_{N+2}, a_{N}, a_N+a_{N+1}+a_{N+2}) .$$
\end{lemma}
We will say we have a triple $a_N,a_{N+1},a_{N+2}$ at level $n$ if
$N=\tau(j_n, \ldots , j_1;1)$.
\begin{proposition} For all positive $n$ and $m$,
$$\delta_n(2m)=\delta_n^1(2m) = \delta_n^2(2m)=\delta_{n}^2(2m)=0.$$
\end{proposition}
\begin{proof}
If
$N=\tau(j_1, \ldots , j_n;1)$ at the $n$th level and if each term in the triple $a_N,a_{N+1},a_{N+2}$ is odd, then all of their their descendants in the next level will still be odd, from the above lemma.
Thus to prove the proposition, we just have to observe that at the initial level $0$, our numbers are
$1,1,1,$ forcing all subsequent elements to be odd.
\end{proof}
\begin{proposition} For any odd number $2n+1>1$, we have
$$\delta_n^3(2n+1) = 9.$$
\end{proposition}
\begin{proof} We first show that if $N=\tau(0^n;1)$, then the corresponding triple is
$(1,1,2n+1)$. The proof is by induction. At the initial zero level, the initial triples $(1,1,1)$ as desired. Then suppose at the $n$th level the triple corresponding to $N=\tau(0^n;1)$ is the desired $(1,1,2n+1)$. Then we have at the next level three triples descending from $(1,1,2n+1),$ namely $(1,1,2n+3), (1, 2n+1, 2n+3)$ and $(2n+1, 1, 2n + 3)$. By the three-fold symmetry of the sequence, this means that
$$\delta_{n+1}(2n+3) \geq 9.$$
Now to show that this number cannot be greater than $9$. Suppose at level $n\geq 1$ there are only three triples of the form $(1,1,2n+1)$. This is true at level one. For any triple $(a_N, a_{N+1},a_{N+2})$ at level $n$ with $a_N$ or $a_{N+1}$ greater than one, then none of this triple's descendants will contribute to $\delta_{n+1}^3(2n+3)$. Then at the next level, we will only pick up three triples $(1,1,2n+3)$. Putting all this together, means that the only triples at level $n$ that will have descendants contributing to $\delta_{n+1}^3(2n+3)$ must be of the form $(1,1,2n+1)$, and each of these three triples will contribute three $(2n+3)$s. Thus we have our result.
\end{proof}
\begin{proposition}
$$\delta_n(m) = \delta_{n+1}^1(m) = \delta_{n+1}^2(m) .$$
\end{proposition}
\begin{proof}
This follows immediately from the construction of the sequence.
\end{proof}
\begin{proposition}
$$\delta_{k+m}(2k+1) = 2^m \delta_k(2k+1).$$
\end{proposition}
\begin{proof}
It is at the $k$th level where $2k+1$ can last occur in a $\delta^3$ term. Hence this follows from the previous proposition, since we have
\begin{eqnarray*}
\delta_{k+m}(2k+1) &=&\delta_{k+m}^1(2k+1) + \delta_{k+m}^2(2k+1)+ \delta_{k+m}^3(2k+1) \\
&=& \delta_{k+m-1}(2k+1) + \delta_{k+m-1}(2k+1) + 0 \\
&=& 2 \delta_{k+m-1}(2k+1).
\end{eqnarray*}
\end{proof}
\begin{proposition}
$$\delta_n(2n+1) = 9 + 2 \delta_{n-1}(2n+1) $$
\end{proposition}
\begin{proof}
We have already seen that $\delta_n^3(2n+1)=9$. Since
$$\delta_n(2n+1) = \delta_n^1(2n+1) + \delta_n^2(2n+1) + \delta_n^3(2n+1)= \delta_n^1(2n+1) + \delta_n^2(2n+1) +9,$$
all we need show is that $\delta_n^1(2n+1) + \delta_n^2(2n+1)= 2 \delta_{n-1}(2n+1)$. But this follows from the fact that each triple $a,b,c$ at the $k-1$st level generates for the $k$th level the triples
$a,b,a+b+a$, or $b,c,a+b+c,$ or $c,a,a+b+c$ and each number appearing in the $n-1$st level appears twice in the $k$th level.
\end{proof}
Of course, it is easy to generate a table of various $\delta_K(n)$. The following is a list of some examples.
\begin{center}
Table $1$
$$\begin{array}{c||c|c|c|c}
(K,n) & \delta_K(n) & \delta_K^1(n) & \delta_K^2(n) & \delta_K^3(n)\\
\hline
(0,1) & 3 & 1 & 1 & 1\\
\hline
(1,1) & 6 & 3 & 3 & 0 \\
\hline
(1,3) &3& 0&0&3\\
\hline
(2,1) & 12 & 6 & 6 & 0 \\
\hline
(2,3) & 6 & 3 & 3 & 0 \\
\hline
(2,5) & 9 & 0 & 0 & 9 \\
\hline(3,1) & 24 & 12 & 12 & 0 \\
\hline
(3,1) & 24&12&12&0\\
\hline
(3,3) & 12 & 6 & 6 & 0\\
\hline
(3,5) & 18 & 9 & 9 & 0\\
\hline
(3,7)& 9&0&0&9 \\
\hline
(3,9)&18&0&0&18\\
\hline
(4,1)&48&24&24&0\\
\hline
(4,3)&24&12&12&0\\
\hline
(4,5)&36&18&18&0 \\
\hline
(4,7)&18&9&9&0 \\
\hline
(4,9)& 45&18&18&9\\
\hline
(4,13)& 36&0&0&36 \\
\hline
(4,15)&18&0&0&18\\
\hline
(4,17)&18&0&0&18\\
\hline
(5,1)&96&48&48&0\\
\hline
(5,3)&48&24&24&0\\
\hline
(5,5)&72&36&36&0\\
\hline
(5,7) & 36&18&18&0\\
\hline
(5,9)&90&45&45&0\\
\hline
(5,11)&9&0&0&9\\
\hline
(5,13)& 72&36&36&0\\
\hline
(5,15) & 36&18&18 &0\\
\hline
(5,17) & 72&18&18&36 \\
\hline
(5,19)& 18&0&0&18 \\
\hline
(5,21) &18&0&0&18 \\
\hline
(5,23)& 18&0&0&18 \\
\hline
(5,25)& 72&0&0&72 \\
\hline
(5,29)&36&0&0&36\\
\hline
(5,31) & 18&0&0&18
\end{array}$$
\end{center}
\section{Paths of a directed graph}\label{sec:Paths of a directed graph}
There is a simple interpretation of Stern's triatomic sequence as the number of paths in a directed graph from three initial vertices. Recall the diagrams in Section~\ref{three}. Starting with a triangle $\triangle$ with vertices $v_1,v_2,v_3$, we systematically constructed a subdivision of $\triangle$ so that at level $n$, we have $3^n$ triangles. It is quite easy to put the structure of a directed graph on this subdivision. Suppose we have one of the subtriangles $\triangle(I)$, where recall $I$ is an $n$-tuple consisting of $0,1$ or $2$. Its vertices are denoted by $v_1(I), v_2(I)$ and $v_3(I)$. Then for level $n+1$, we add one more vertex, denoted by $v_1(I) + v_2(I)+v_3(I)$ and three directed paths, one from each $v_1(I)$ to the new vertex. Finally, note that there are no directed paths between any of the three initial vertices. Recall
our earlier notation of writing any positive integer $N$ as
$$N=\tau(j_1, j_{2}, \ldots j_n;k)$$
if
$$N=3(1 + 3 + 3^2 + + 3^{n-1} ) + j_{1}3^n + j_{2}3^{n -1} + \cdots + j_n3 + k,$$
where each $j_i$ is zero, one or two and $l$ is chosen to be as large as possible from the set $\{1,2,3\}$.
\begin{theorem}
For any positive integer $N$, the number $a_N$ is the number of paths from $v_1, v_2$ and $v_3$ to the $l$th vertex of the $\triangle(j_1, \ldots , j_n),$ save for when the corresponding vertex of $\triangle(j_n, \ldots , j_1)$ is one of the initial $v_1, v_2$ or $v_3$.
\end{theorem}
The proof is straightforward.
\section{Questions}
The Farey map is only one type of multidimensional continued fraction. (For background on multidimensional continued fractions, see Schweiger \cite{Schweiger1}). Recently, Dasaratha et al. \cite{DasarathaFlapanGarrityLeeMihailaNeumann-Chun-Peluse-Stoffregen1} produced a family of multidimensional continued fractions which includes many previously well-known algorithms, though not the Farey map. Using this family, Dasaratha et al. \cite{Dasaratha et al 12 d} study analogs of Stern's diatomic sequence. Independently, Goldberg \cite{Goldberg12} has started finding analogs for the Monkemayer map. It would also be interesting to find the Stern analogs in the language of Lagarias \cite{Lagarias93}.
Also, there are many properties of Stern's diatomic sequences whose analogs for the Farey map have yet to be discovered. Probably the most interesting would be to find the analog of the link between Stern's sequence and the number of hyperbinary representations there are for positive integers, as discussed by Northshield \cite{Northshield1}.
Another interesting problem is to attempt a multidimensional analog of the recently found link between the Tower of Hanoi graph and Stern's diatomic sequence by Hinz, Klav\v{z}ar, Milutinovi\'{c}, Parisse and Petr \cite{AH}.
Finally, it would be interesting to extend the polynomial analogs of Stern's diatomic sequence (as in the work of Dilcher and Stokarsky \cite{Dilcher-Stokarsky07, Dilcher-Stokarsky09}, of Klav\v{z}ar, Milutinovi\'{c}
and Petr \cite{Klavzar-Milutinovic-Petr07}, of Ulas \cite{Ulas-Ulas11, Ulas12}, of Vargas \cite{Vargas12}, and of Allouche and Mend\`{e}s France \cite{Allouche-Mendes France12}) to finding polynomial analogs for triatomic sequences.
\section{Acknowledgments}
We would like to thank the referee for major help in improving the exposition of this paper, and catching a number of errors.
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\bigskip
\hrule
\bigskip
\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B83; Secondary 11A55, 11J70, 40A99.
\noindent \emph{Keywords: } Stern's diatomic sequence,
multidimensional continued fractions.
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\noindent (Concerned with sequences
\seqnum{A002487} and \seqnum{A228925}.)
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Received February 10 2013;
revised versions received February 21 2013; September 5 2013;
September 8 2013.
Published in {\it Journal of Integer Sequences}, September 8 2013.
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