Introducing supersymmetric frieze patterns and linear difference operators

Abstract

We introduce a supersymmetric analog of the classical Coxeter frieze patterns. Our approach is based on the relation with linear difference operators. We define supersymmetric analogs of linear difference operators called Hill’s operators. The space of these “superfriezes” is an algebraic supervariety, isomorphic to the space of supersymmetric second order difference equations, called Hill’s equations.

Appendices

Address: Khabarovsk Division, Institute of Applied Mathematics, Russian Academy of Sciences, 54 Dzerzhinsky Street, Khabarovsk 680000, Russia, E-mail: ustinov@iam.khv.ru

Appendix 1: Elements of superalgebra

To make the paper self-contained, we briefly describe several elementary notions of superalgebra and supergeometry used above. For more details, we refer to the classical sources [2, 6, 19, 21, 22].

1.1 Supercommutative algebras

Let Latin letters denote even variables, and Greek letters the odd ones. Consider algebras of polynomials $\mathbb {K}[x_1,\ldots ,x_n,\xi _1,\ldots ,\xi _k]$, where $\mathbb {K}=\mathbb {R},\mathbb {C}$, or some other supercommutative ring, and where $x_i$ are standard commuting variables, while the odd variables $\xi _i$ commute with $x_i$ and anticommute with each other:

$$\begin{aligned} \xi _i\xi _j=-\xi _j\xi _i, \end{aligned}$$

for all i, j; in particular, $\xi _i^2=0$. Every supercommutative algebra is a quotient of a polynomial algebra by some ideal. Every supercommutative algebra is the algebra of regular functions on an algebraic supervariety (which can be taken for a definition of the latter notion). Every Lie superalgebra is the algebra of derivations of a supercommutative algebra, for instance, vector fields are derivations of the algebra of regular functions on an algebraic supervariety.

An example of supercommutative algebra is the Grassmann algebra of differential forms on a vector space. Let $(x_1,\ldots ,x_n)$ be coordinates, and $(dx_1,\ldots ,dx_n)$ their differentials, one replaces all the differentials $dx_i$ by the odd variables $\xi _i$, to obtain an isomorphic algebra.

We often need to calculate rational functions with odd variables. The main ingredient is the obvious formula $(1+\xi )^{-1}=1-\xi$. For instance, we have:

$$\begin{aligned} \frac{y}{x+\xi }=\frac{y}{x(1+\xi /x)}=\frac{y}{x}-\xi \frac{y}{x^2}. \end{aligned}$$

1.2 The supergroup $\mathrm {OSp}(1|2)$

The supergroup $\mathrm {OSp}(1|2)$ is isomorphic to the supergroup of linear symplectic transformations of the 2|1-dimensional space equipped with the symplectic form

$$\begin{aligned} \omega =dp\wedge {}dq+\frac{1}{2}d\tau \wedge {}d\tau , \end{aligned}$$

where $p,q,\tau$ are linear coordinates.

Let $\mathcal {R}=\mathcal {R}_{\bar{0}}\oplus \mathcal {R}_{\bar{1}}$ be a commutative ring. The set of $\mathcal {R}$-points of the supergroup $\mathrm {OSp}(1|2)$ is the following 3|2-dimensional supergroup of matrices with entries in $\mathcal {R}$:

$$\begin{aligned} \left( \begin{array}{cc|c} a&{}\quad b&{}\quad \gamma \\ c&{}\quad d&{}\quad \delta \\ \hline \alpha &{}\quad \beta &{}\quad e \end{array} \right) \quad \hbox {such that} \quad \left\{ \begin{array}{l} ad-bc=1-\alpha \beta ,\\ e=1+\alpha \beta ,\\ -a\delta +c\gamma =\alpha \\ -b\delta +d\gamma =\beta , \end{array} \right. \end{aligned}$$

where $a,b,c,d,e\in \mathcal {R}_{\bar{0}}$, and $\alpha ,\beta ,\gamma ,\delta \in \mathcal {R}_{\bar{1}}$. For properties and applications of this supergroup, see [15, 21]. Note that the above relations also imply:

$$\begin{aligned} \gamma =a\beta -b\alpha , \quad \delta =c\beta -d\alpha , \end{aligned}$$

and $\alpha \beta =\gamma \delta$.

1.3 Left-invariant vector fields on $R^{1|1}$ and supersymmetric linear differential operators

Consider the space $\mathbb {R}^{1|1}$ with linear coordinates $(x,\xi )$. We understand the algebra of algebraic functions on this space as the algebra of polynomials in one even and one odd variables:

$$\begin{aligned} F(x,\xi )=F_0(x)+\xi {}F_1(x), \end{aligned}$$

where $F_0$ and $F_1$ are usual polynomials in x.

The following two vector fields, characterized by Shander’s superversion of the rectifiability of vector fields theorem [33]

$$\begin{aligned} X=\frac{\partial }{\partial {}x}, \quad D=\frac{\partial }{\partial {}\xi }-\xi \frac{\partial }{\partial {}x} \end{aligned}$$

are important in superalgebra and supergeometry. These vector fields are left-invariant with respect to the supergroup structure on $\mathbb {R}^{1|1}$ given by the following multiplication of $\mathcal {R}$-points:

$$\begin{aligned} (r,\lambda )\cdot (s,\mu )=(r+s+\lambda \mu ,\,\lambda +\mu ). \end{aligned}$$

Moreover, X and D are characterized (up to a constant factor) by the property of left-invariance, as the only even and odd left-invariant vector fields on $\mathbb {R}^{1|1}$, respectively.

The vector fields X and D form a 1|1-dimensional Lie superalgebra since

$$\begin{aligned} D^2=\frac{1}{2}[D,D]=-X, \end{aligned}$$

and $[X,D]=0$, with one odd generator D.

The space $\mathbb {R}^{1|1}$ equipped with the vector field D is often called by physicists the 1|1-dimensional “superspacetime”. A supersymmetric differential operator on $\mathbb {R}^{1|1}$ is an operator that can be expressed as a polynomial in D.

Appendix 2: Supercontinuants (by Alexey Ustinov)

Address: Khabarovsk Division, Institute of Applied Mathematics, Russian Academy of Sciences, 54 Dzerzhinsky Street, Khabarovsk 680000, Russia, E-mail: ustinov@iam.khv.ru

This Appendix gives a solution to Problem 1: determine the formula for the entries of a superfrieze.

Let $\mathcal {R}=\mathcal {R}_{\bar{0}}\oplus \mathcal {R}_{\bar{1}}$ be an arbitrary supercommutative ring, and the sequences $\{v_i\}$, $\{w_i\}$, with $v_i\in \mathcal {R}_{\bar{0}}$, $w_i\in \mathcal {R}_{\bar{1}}$, be defined by the initial conditions $v_{-1}=0$, $v_0=1$, $w_0=0$ and the recurrence relation

$$\begin{aligned} v_i=a_iv_{i-1}-v_{i-2}-\beta _iw_{i-1},\quad w_i=w_{i-1}+\beta _iv_{i-1}\quad (i\in \mathbb {Z}). \end{aligned}$$

(4.1)

In particular,

$$\begin{aligned} v_1= & {} a_1,\quad v_2=a_1a_2-1+\beta _1\beta _2,\quad v_3=a_1a_2a_3-a_1-a_3+a_1\beta _2\beta _3+a_3\beta _1\beta _2+\beta _1\beta _3; \\ w_1= & {} \beta _1,\quad w_2=a_1\beta _2+\beta _1,\quad w_3= a_1a_2\beta _3+a_1\beta _2+\beta _1\beta _2\beta _3+\beta _1-\beta _3. \end{aligned}$$

The problem is to express $v_n$, $w_n$ in terms of $a_1$, ..., $a_n$ and $\beta _1$, ..., $\beta _n$. Such expression will be called supercontinuants. (For the properties of the classical continuants, see [14].)

We define two sequences of supercontinuants

$$\begin{aligned} \{K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|{\begin{matrix} a_2\\ {\begin{matrix} \beta _2 &{} \beta _2 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_n\\ {\begin{matrix} \beta _n &{} \beta _n \end{matrix}} \end{matrix}}\bigr )\}\quad \text {and} \quad \{K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}|\beta _n\bigr )\} \end{aligned}$$

by the initial conditions $K()=1$, $K({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}})=a_1$, $K(\beta _1)=\beta _1$ and the recurrence relations

$$\begin{aligned} K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_n\\ {\begin{matrix} \beta _n &{} \beta _n \end{matrix}} \end{matrix}}\bigr )= & {} a_nK\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}\bigr )-K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-2}\\ {\begin{matrix} \beta _{n-2} &{} \beta _{n-2} \end{matrix}} \end{matrix}}\bigr )\nonumber \\&-\beta _nK\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-2}\\ {\begin{matrix} \beta _{n-2} &{} \beta _{n-2} \end{matrix}} \end{matrix}}|\beta _{n-1}\bigr ),\nonumber \\ K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}|\beta _n\bigr )= & {} \beta _nK\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}\bigr )\nonumber \\&+K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-2}\\ {\begin{matrix} \beta _{n-2} &{} \beta _{n-2} \end{matrix}} \end{matrix}}|\beta _{n-1}\bigr ). \end{aligned}$$

(4.2)

From (4.1) and (4.2) it easily follows that

$$\begin{aligned} v_n=K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_n\\ {\begin{matrix} \beta _n &{} \beta _n \end{matrix}} \end{matrix}}\bigr ), \quad w_n=K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}|\beta _n\bigr ). \end{aligned}$$

The classical continuants $K(a_1,\ldots ,a_n)$, corresponding to reduced regular continued fractions

$$\begin{aligned} a_1-\frac{1}{a_2-{_{\ddots \,\displaystyle {-\frac{1}{a_n}}}}}, \end{aligned}$$

are defined by

$$\begin{aligned} K()=1,\quad K(a_1)=a_1,\quad K(a_1,\ldots ,a_n)=a_nK(a_1,\ldots ,a_{n-1})-K(a_1,\ldots ,a_{n-2}). \end{aligned}$$

There is Euler’s rule which allows one to write down all summands of $K(a_1,\ldots ,a_n)$: starting with the product $a_1 a_2 \ldots a_n$, we strike out adjacent pairs $a_ia_{i+1}$ in all possible ways. If a pair $a_ia_{i+1}$ is struck out, then it must be replaced by $-1$. We can represent Euler’s rule graphically by constructing all “Morse code” sequences of dots and dashes having length n, where each dot contributes 1 to the length and each dash contributes 2. For example $K(a_1,a_2,a_3,a_4)$ consists of the following summands:

By analogy with Euler’s rule, we can construct a similar rule for calculation of supercontinuants.

Theorem 4.1

The summands of $K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|{\begin{matrix} a_2\\ {\begin{matrix} \beta _2 &{} \beta _2 \end{matrix}} \end{matrix}}|\ldots \bigr )$ can be obtained from the product $\beta _1\beta _1\beta _2\beta _2\ldots$ by the following rule: we strike out adjacent pairs and adjacent 4-tuples $\beta _i\beta _i\beta _{i+1}\beta _{i+1}$ in all possible ways; for deleted pairs and 4-tuples we make the substitutions $\beta _i\beta _i\rightarrow a_i$, $\beta _i \beta _{i+1}\rightarrow 1$, $\beta _i\beta _i\beta _{i+1}\beta _{i+1}\rightarrow -1.$

This rule can be represented graphically as well. To each monomial there corresponds a sequence of total length 2n (or $2n-1$) consisting of dots (of the length one), dashes (of the length two) and long dashes (of the length four). For example, the monomials of $K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|{\begin{matrix} a_2\\ {\begin{matrix} \beta _2 &{} \beta _2 \end{matrix}} \end{matrix}}|\beta _{3}\bigr )$ can be obtained from the product $\beta _1\beta _1 \beta _2\beta _2\beta _{3}$ as follows:

Let us note that the odd variables anticommute with each other. In particular, $\beta _i^2=0$, and in each pair $\beta _i\beta _i$ at least one variable must be struck out. Supercontinuants become the usual continuants if all odd variables are replaced by zeros.

Supercontinuants can be expressed as determinants.

Theorem 4.2

$$\begin{aligned}&K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_n\\ {\begin{matrix} \beta _n &{} \beta _n \end{matrix}} \end{matrix}}\bigr )\!=\!\begin{vmatrix} a_1&\quad -1+\beta _1\beta _2&\quad \beta _1\beta _3&\quad \cdots&\quad \beta _1\beta _{n-1}&\quad \beta _1\beta _n \\ -1&\quad a_2&\quad -1+\beta _2\beta _3&\quad \cdots&\quad \beta _2\beta _{n-1}&\quad \beta _2\beta _n \\ 0&\quad -1&\quad a_3&\quad \cdots&\quad \beta _3\beta _{n-1}&\quad \beta _3\beta _n \\ \cdots&\quad \cdots&\quad \cdots&\quad \cdots&\quad \cdots&\quad \cdots \\ 0&\quad \cdots&\quad 0&\quad -1&\quad a_{n-1}&\quad -1+\beta _{n-1}\beta _n \\ 0&\quad 0&\quad \cdots&\quad 0&\quad -1&\quad a_n \\ \end{vmatrix},\nonumber \\&K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}|\beta _n\bigr )\nonumber \\&\quad =\begin{vmatrix} a_1&\quad -1+\beta _1\beta _2&\quad \beta _1\beta _3&\quad \cdots&\quad \beta _1\beta _{n-1}&\quad \beta _1\\ -1&\quad a_2&\quad -1+\beta _2\beta _3&\quad \cdots&\quad \beta _2\beta _{n-1}&\quad \beta _2 \\ 0&\quad -1&\quad a_3&\quad \cdots&\quad \beta _3\beta _{n-1}&\quad \beta _3\\ \cdots&\quad \cdots&\quad \cdots&\quad \cdots&\quad \cdots&\quad \cdots \\ 0&\quad \cdots&\quad -1&\quad a_{n-2}&\quad -1+\beta _{n-2}\beta _{n-1}&\quad \beta _{n-2}\\ 0&\quad \cdots&\quad 0&\quad -1&\quad a_{n-1}&\quad \beta _{n-1}\\ 0&\quad 0&\quad \cdots&\quad 0&\quad -1&\quad \beta _n \\ \end{vmatrix}. \end{aligned}$$

(4.3)

The second determinant in Theorem 4.2 is well-defined because odd variables occupy only one column. The proofs of Theorems 4.1 and 4.2 follow by induction from recurrence relations (4.2), and we do not dwell on them.

The supercontinuants of the form $K\bigl (\beta _1|{\begin{matrix} a_2\\ {\begin{matrix} \beta _2 &{} \beta _2 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}|\beta _n\bigr )$ also may be defined by the rule from the Theorem 4.1. For example

$$\begin{aligned} K(\beta _1|\beta _2)=\beta _1\beta _2+1,\quad K\bigl (\beta _1|{\begin{matrix} a_2\\ {\begin{matrix} \beta _2 &{} \beta _2 \end{matrix}} \end{matrix}}|\beta _3\bigr )=a_2\beta _1\beta _3+\beta _1\beta _2+\beta _2\beta _3+1. \end{aligned}$$

These supercontinuants can be represented in terms of determinants as well (we assume that the determinant is expanded in the first column, and the same rule is applied to all determinants of smaller matrices).

Theorem 4.3

The supercontinuants $K\bigl (\beta _1|{\begin{matrix} a_2\\ {\begin{matrix} \beta _2 &{} \beta _2 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}|\beta _n\bigr )$ satisfy the recurrence relation

$$\begin{aligned} \begin{array}{rcl} K\bigl (\beta _1|{\begin{matrix} a_2\\ {\begin{matrix} \beta _2 &{} \beta _2 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}|\beta _n\bigr ) &{}=&{} -\beta _nK\bigl (\beta _1|{\begin{matrix} a_2\\ {\begin{matrix} \beta _2 &{} \beta _2 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}\bigr )\\ &{}&{}+\,K\bigl (\beta _1|{\begin{matrix} a_2\\ {\begin{matrix} \beta _2 &{} \beta _2 \end{matrix}} \end{matrix}}|\ldots |\beta _{n-1}\bigr ) \end{array} \end{aligned}$$

(4.4)

and can be expressed in the following form:

$$\begin{aligned} K\bigl (\beta _1|{\begin{matrix} a_2\\ {\begin{matrix} \beta _2 &{} \beta _2 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}|\beta _n\bigr )=\begin{vmatrix} \beta _1&\quad \beta _2&\quad \beta _3&\quad \cdots&\quad \beta _{n-1}&\quad 1\\ -1&\quad a_2&\quad -1+\beta _2\beta _3&\quad \cdots&\quad \beta _2\beta _{n-1}&\quad \beta _2 \\ 0&\quad -1&\quad a_3&\quad \cdots&\quad \beta _3\beta _{n-1}&\quad \beta _3\\ \cdots&\quad \cdots&\quad \cdots&\quad \cdots&\quad \cdots&\quad \cdots \\ 0&\quad \cdots&\quad -1&\quad a_{n-2}&\quad -1+\beta _{n-2}\beta _{n-1}&\quad \beta _{n-2}\\ 0&\quad \cdots&\quad 0&\quad -1&\quad a_{n-1}&\quad \beta _{n-1}\\ 0&\quad 0&\quad \cdots&\quad 0&\quad -1&\quad \beta _n \\ \end{vmatrix}. \end{aligned}$$

The proof of formula (4.4) is an application of the rule from Theorem 4.1. The determinant formula follows by induction from the recurrence relation (4.4).

Finally, the even supercontinuants $K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_n\\ {\begin{matrix} \beta _n &{} \beta _n \end{matrix}} \end{matrix}}\bigr )$ can be also expressed as Berezinians. Recall that the Berezinian of the matrix

$$\begin{aligned} \mathrm {Ber} \begin{pmatrix} A&{}B\\ C&{}D\\ \end{pmatrix}, \end{aligned}$$

where A and D have even entries, and B and C have odd entries, is given by (see, e.g., [2]):

$$\begin{aligned} \det (A-BD^{-1}C)\det (D)^{-1}. \end{aligned}$$

(4.5)

Theorem 4.4

$$\begin{aligned} K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_n\\ {\begin{matrix} \beta _n &{} \beta _n \end{matrix}} \end{matrix}}\bigr )=\mathrm {Ber}\begin{pmatrix} A&{}B\\ C&{}D\\ \end{pmatrix}, \end{aligned}$$

where

Theorem 4.4 is direct corollary of (4.4) and (4.5).

It follows from recurrence relations (4.2) and (4.4) that the number of terms in supercontinuants $K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_n\\ {\begin{matrix} \beta _n &{} \beta _n \end{matrix}} \end{matrix}}\bigr )$, $K\bigl ({\begin{matrix} a_1\\ {\begin{matrix} \beta _1 &{} \beta _1 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}|\beta _n\bigr )$ and $K\bigl (\beta _1|{\begin{matrix} a_2\\ {\begin{matrix} \beta _2 &{} \beta _2 \end{matrix}} \end{matrix}}|\ldots |{\begin{matrix} a_{n-1}\\ {\begin{matrix} \beta _{n-1} &{} \beta _{n-1} \end{matrix}} \end{matrix}}|\beta _n\bigr )$ coincide respectively with the sequences (see [34])

$$\begin{aligned} A077998&{:}\,1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, 4004, \ldots \\ A006054&{:}\,1, 2, 5, 11, 25, 56, 126, 283, 636, 1429, 3211, \ldots \\ A052534&{:}\,1, 2, 4, 9, 20, 45, 101, 227, 510, 1146, 2575, \ldots \end{aligned}$$