Modular Periodicity of the Euler Numbers and a Sequence by Arnold

Abstract

For any positive integer q, the sequence of the Euler up/down numbers reduced modulo q was proved to be ultimately periodic by Knuth and Buckholtz. Based on computer simulations, we state for each value of q precise conjectures for the minimal period and for the position at which the sequence starts being periodic. When q is a power of 2, a sequence defined by Arnold appears, and we formulate a conjecture for a simple computation of this sequence.

1 Introduction

The sequence of Euler up/down numbers $(E_n)_{n\ge 0}$ is the sequence with exponential generating series

$$\begin{aligned} \sum _{n=0}^{\infty } \frac{E_n}{n!} x^n = \sec x + \tan x. \end{aligned}$$

(1)

It is referenced as sequence A000111 in the On-Line Encyclopedia of Integer Sequences (Sloane 2017) and its first terms are

$$\begin{aligned} 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765,\ldots \end{aligned}$$

The numbers $E_n$ were shown by André (1879) to count up/down permutations on n elements (see Sect. 3).

Knuth and Buckholtz (1967) proved that for any integer $q\ge 1$, the sequence $(E_n \mod q)_{n\ge 0}$ is ultimately periodic. For any $q \ge 1$ we define:

• s(q) to be the minimum number of terms one needs to delete from the sequence $(E_n \mod q)_{n \ge 0}$ to make it periodic;

• d(q) to be the smallest period of the sequence $(E_n \mod q)_{n \ge s(q)}$.

For example, the sequence $(E_n \mod 3)$ starts with

$$\begin{aligned} 1,1,1,2,2,1,1,2,2,1,1,2,2,\ldots \end{aligned}$$

so one might expect to have $s(3)=1$ and $d(3)=4$. Clearly $s(1)=0$ and $d(1)=1$. In the remainder of this paper, we formulate precise conjectures for the values of s(q) and d(q) for any $q\ge 2$.

Organisation of the paper In Sect. 2 we reduce the problem to the case when q is a prime power and we conjecture the values of s(q) and d(q) when q is an odd prime power. In Sect. 3 we conjecture the values of s(q) and d(q) when q is a power of 2, after having introduced the Entringer numbers and a sequence defined by Arnold describing the 2-adic valuation of the Entringer numbers. In Sect. 4, we provide a simple construction which conjecturally yields the Arnold sequence.

2 Case When q is Not a Power of 2

The following lemma implies that it suffices to know the values of s(q) and d(q) when q is a prime power in order to know the values of s(q) and d(q) for any $q \ge 2$.

Lemma 1

Fix $q \ge 2$ and write its prime number decomposition as

$$\begin{aligned} q=\prod _{i=1}^k p_i^{\alpha _i}, \end{aligned}$$

(2)

where $k\ge 1,$ $p_1,\ldots ,p_k$ are distinct prime numbers and $\alpha _1,\ldots ,\alpha _k$ are positive integers. Then

$$\begin{aligned} s(q)= & {} \max _{1 \le i \le k} s(p_i^{\alpha _i}) \end{aligned}$$

(3)

$$\begin{aligned} d(q)= & {} {{\mathrm{lcm}}}(d(p_1^{\alpha _1}),\ldots ,d(p_k^{\alpha _k})). \end{aligned}$$

(4)

The proof is elementary and uses the Chinese remainder theorem.

When q is an odd prime power, Knuth and Buckholtz (1967) found the following:

Theorem 2

(Knuth and Buckholtz 1967) Let p be an odd prime number.

• (1) If $p \equiv 1 \mod 4,$ then

  $$\begin{aligned} d(p)=p-1. \end{aligned}$$

• (2) If $p \equiv 3 \mod 4,$ then

  $$\begin{aligned} d(p)=2p-2. \end{aligned}$$

• (3) For any $k\ge 1,$

  $$\begin{aligned} s(p^k)\le k. \end{aligned}$$

• (4) For any $k\ge 2,$

  $$\begin{aligned} d(p^k) | p^{k-1} d(p). \end{aligned}$$

We conjecture the following for the exact values of s(q) and d(q) when q is an odd prime power:

Conjecture 1

Let p be an odd prime number.

• (1) For any $k\ge 1,$

  $$\begin{aligned} s(p^k)=k. \end{aligned}$$

• (2) For any $k\ge 2,$

  $$\begin{aligned} d(p^k)=p^{k-1} d(p). \end{aligned}$$

Conjecture 1 is supported by Mathematica simulations done for all odd prime powers $q<1000$.

3 Entringer Numbers and Case When q is a Power of 2

Formulating a conjecture analogous to Conjecture 1 for powers of 2 requires to define, following Arnold (1991), a sequence describing the behavior of the 2-adic valuation of the Entringer numbers.

3.1 The Seidel–Entringer–Arnold Triangle

The Entringer numbers are a refined version of the Euler numbers, enumerating some subsets of up/down permutations. For any $n\ge 0$, a permutation $\sigma \in \mathcal S_n$ is called up/down if for any $2\le i \le n$, we have $\sigma (i-1)<\sigma (i)$ (resp. $\sigma (i-1)>\sigma (i)$) if i is even (resp. i is odd). André (1879) showed that the number of up/down permutations on n elements is $E_n$. For any $1 \le i \le n$, the Entringer number $e_{n,i}$ is defined to be the number of up/down permutations $\sigma \in \mathcal S_n$ such that $\sigma (n)=i$. The Entringer numbers are usually displayed in a triangular array called the Seidel–Entringer–Arnold triangle, where the numbers $(e_{n,i})_{1 \le i\le n}$ appear from left to right on the n-th line (see Fig. 1).

Fig. 1

First five lines of the Seidel–Entringer–Arnold triangle

The Entringer numbers can be computed using the following recurrence formula (see for example Stanley 2012). For any $n\ge 2$ and for any $1\le i \le n$, we have

$$\begin{aligned} e_{n,i}= {\left\{ \begin{array}{ll} \sum _{j<i} e_{n-1,j} &{}\text {if n is even} \\ \sum _{j\ge i} e_{n-1,j} &{}\text {if n is odd}. \end{array}\right. } \end{aligned}$$

(5)

3.2 Arnold’s Sequence

Replacing each entry of the Seidel–Entringer–Arnold triangle by its 2-adic valuation, we obtain an infinite triangle denoted by T (see Fig. 2).

Fig. 2

First five lines of the triangle T of 2-adic valuations of the Entringer numbers

We read this triangle T diagonal by diagonal, with diagonals parallel to the left boundary. For any $i\ge 1$, denote by $D_i$ the i-th diagonal of the triangle T parallel to the left boundary. For example $D_1$ starts with $0,\infty ,0,\infty ,0,\ldots$ For any $i\ge 1$, denote by $m_i$ the minimum entry of diagonal $D_i$. Arnold (1991) observed that the further away one moves from the left boundary, the higher the 2-adic valuation of the Entringer numbers becomes. In particular, he observed (without proof) that the sequence $(m_i)_{i\ge 1}$ was weakly increasing to infinity. He defined the following sequence: for any $k \ge 1$,

$$\begin{aligned} u_k:=\max \left\{ i\ge 1 | m_i < k \right\} . \end{aligned}$$

In other words, $u_k$ is the number of diagonals containing at least one entry that is not zero modulo $2^k$. The sequence $(u_k)_{k\ge 1}$ is referenced as the sequence A108039 in the On-Line Encyclopedia of Integer Sequences (Sloane 2017) and its first few terms are given in Table 1.

Table 1 The first few values of $u_k$

Note that the first few terms given by Arnold were incorrect, because the entry 4 appeared four times, whereas it should be appearing only three times. We also remark that we cannot define any sequence analogous to $(u_k)$ when studying the p-adic valuations of the Entringer numbers for odd primes p. Indeed, the p-adic valuation 0 seems to appear in diagonals of arbitrarily high index.

3.3 Case When q is a Power of 2

Using the sequence $(u_k)_{k\ge 1}$, we formulate the following conjecture for s(q) and d(q) when q is a power of 2:

Conjecture 2

For any $k\ge 1,$ we have

$$\begin{aligned} s(2^k)=u_k. \end{aligned}$$

(6)

Furthermore,  if $k\ge 1$ and $k\ne 2,$ we have

$$\begin{aligned} d(2^k)= 2^k. \end{aligned}$$

(7)

Finally,  we have $d(4)=2$.

Numerical simulations performed on Mathematica for $k\le 12$ support Conjecture 2.

4 Construction of Arnold’s Sequence

In this section we provide a construction which conjecturally yields Arnold’s sequence $(u_k)_{k\ge 1}$.

We denote by $\mathbb Z_+$ the set of nonnegative integers and we denote by

$$\begin{aligned} S:=\bigsqcup _{d\ge 1}\mathbb Z_+^d \end{aligned}$$

the set of all finite sequences of nonnegative integers. We define a map $f:S\rightarrow S$, which maps each $\mathbb Z_+^d$ to $\mathbb Z_+^{2d}$, as follows. Fix $\underline{x}=(x_1,\ldots ,x_d)\in S$. If all the $x_i$’s are equal to $x_d$, we set

$$\begin{aligned} f(\underline{x})=(x_d,\ldots ,x_d,2x_d,\ldots ,2x_d), \end{aligned}$$

where $x_d$ and $2x_d$ both appear d times on the right-hand side. Otherwise, define

$$\begin{aligned} s:=\max \left\{ 1 \le i \le d-1 | x_i \ne x_d \right\} \end{aligned}$$

and set

$$\begin{aligned} f(\underline{x})=(x_1,\ldots ,x_d,x_1+x_d,\ldots ,x_{s-1}+x_d,2x_d,\ldots ,2x_d), \end{aligned}$$

where $2x_d$ appears $d-s+1$ times on the right-hand side. For example, we have

$$\begin{aligned} f((2,4,4,4))=(2,4,4,4,8,8,8,8) \end{aligned}$$

(8)

and

$$\begin{aligned} f((2,4,4,4,8,8,8,8))=(2,4,4,4,8,8,8,8,10,12,12,16,16,16,16,16). \end{aligned}$$

(9)

By iterating this function f indefinitely, one produces an infinite sequence:

Lemma 3

Fix $d\ge 1$ and $\underline{x}\in \mathbb Z_+^d$. There exists a unique (infinite) sequence $(X_k)_{k\ge 1}$ such that for any $k\ge 1$ and for any $n\ge \log _2 (k/d),$ $X_k$ is the k-th term of the finite sequence $f^n(\underline{x})$.

This infinite sequence is called the f-transform of $\underline{x}$. The lemma follows from the observation that for any $\ell \ge 1$ and for any $\underline{y}\in \mathbb Z_+^\ell$, $\underline{y}$ and $f(\underline{y})$ have the same first $\ell$ terms.

We can now formulate a conjecture about the construction of the sequence $(u_k)_{k\ge 1}$:

Conjecture 3

Arnold’s sequence $(u_k)_{k\ge 1}$ is the f-transform of the quadruple (2, 4, 4, 4).

Conjecture 3 is supported by the estimation on Mathematica of $u_k$ for every $k\le 512$.