DUAL NUMBERS, WEIGHTED QUIVERS, AND EXTENDED SOMOS AND GALE-ROBINSON SEQUENCES. To Alexandre Alexandrovich Kirillov on his 3 4 th anniversary

DUAL NUMBERS, WEIGHTED QUIVERS, AND EXTENDED SOMOS AND GALE-ROBINSON SEQUENCES VALENTIN OVSIENKO AND SERGE TABACHNIKOV Abstract. We investigate a general method that allows one to construct new integer seuences extending existing ones. We aly this method to the classic Somos-4 and Somos-5, and the Gale-Robinson seuences, as well as to more general class of seuences introduced by Fordy and Marsh, and roduce a great number of new seuences. The method is based on the notion of weighted uiver, a uiver with a Z-valued function on the set of vertices that obeys very secial rules of mutation. To Alexandre Alexandrovich Kirillov on his 3 4 th anniversary 1. Introduction A dual number is a air of real numbers a and b, written in the form a + bε, subject to the condition that ε 2 = 0. Dual numbers form a commutative algebra, they were introduced by Clifford in 1873, and since then have found alications in geometry and mathematical hysics. For examle, according to E. Study, the sace of oriented lines in R 3 is the unit shere in the 3-dimensional sace over dual numbers [15]. Surrisingly, dual numbers are not freuent guests in number theory and combinatorics. In this aer, we will use dual numbers to construct a large family of integer seuences. Let (a n ) n N be an integer seuence defined by some recurrence and initial conditions. We will consider a air of seuences, (a n ) n N and (b n ) n N, organized as a seuence of dual numbers: (1) A n := a n + b n ε. We assume that (A n ) n N satisfies either exactly the same recurrence as (a n ) n N, or its certain deformation. We furthermore choose the same initial conditions for (a n ) n N and arbitrary initial conditions for (b n ) n N. We show that, in many interesting cases, the seuence (b n ) n N, defined in this way, is an integer seuence. This method was suggested in [13], and tested on the Somos-4 seuence, roducing new integer seuences. 1.1. Somos, Gale-Robinson and beyond. Let us give a brief overview of the integer seuences that we will consider. The Somos-4 and Somos-5 seuences are the seuences of integers, (a n ) n N, defined by the recurrences: a n+4 a n = a n+3 a n+1 + a 2 n+2 and a n+5 a n = a n+4 a n+1 + a n+3 a n+2, and the initial conditions: a 1 = a 2 = a 3 = a 4 = 1 and a 1 = a 2 = a 3 = a 4 = a 5 = 1, resectively. These seuences were discovered by Michael Somos in 80 s, and they are discrete analogs of ellitic functions. Their integrality, observed and conjectured by Somos, was later roved by several authors; for a historic account see [7]. A more general class of integer seuences generalizing the Somos seuences, called the three term Gale-Robinson seuences, are defined by the recurrences: (2) a n+n a n = a n+n r a n+r + a n+n s a n+s,

2 VALENTIN OVSIENKO AND SERGE TABACHNIKOV where 1 r < s N 2, and the initial conditions (a 1,..., a N ) = (1,..., 1). Their integrality was roved in [4]. A combinatorial roof was then given in [1] and [14], a roof that exlicitly uses uiver mutations and cluster algebras was resented in [6]. A large class of integer seuences generalizing those of Gale-Robinson was introduced in [6] (see also [10]). The recurrence is of the general form (3) a n+n a n = P (a n+n 1,..., a n+1 ), where the olynomial P is a sum of two monomials: P = P 1 + P 2. The initial conditions are those of Gale-Robinson. A simle examle of such seuences is: a n+4 a n = a n+3 a n+1 + a n+2, with arbitrary ositive integers,. Note that this articular seuence was already considered in [7], where their integrality was clamed. The method of [6] is based on the Fomin-Zelevinsky Laurent henomenon [4] and on the notion of eriod 1 uiver, i.e., a uiver that rotates under mutations. 1.2. Extensions. The same arguments show that, for A n = a n + b n ε satisfying the recurrence A n+n A n = P (A n+n 1,..., A n+1 ), where P is as in [6], and (a 1,..., a N ) = (1,..., 1), the seuence (b n ) n N is integer for an arbitrary choice of integral initial conditions (b 1,..., b N ). In fact, this is a direct conseuence of the classic Laurent henomenon. Note that the recurrence for (b n ) n N obtained in this way, is nothing other than the linearization of (3). This linearization rocedure already rovides a large number of new integer seuences. We will also consider the recurrences of the following form: (4) A n+n A n = P 1 + P 2 (1 + wε) and A n+n A n = P 1 (1 + wε) + P 2, where w is an arbitrary integer, and P i stands for P i (A n+n 1,..., A n+1 ), i = 1, 2. In this case, (b n ) n N satisfies a non-linear recurrence. More recisely, the recurrence for (b n ) n N is given by an affine function with olynomial in (a n ) n N coefficients. We show, in articular, that these non-linear extensions of the Gale-Robinson seuences are always integer. We give a sufficient condition for the corresonding eriod 1 uiver that guaranties that the extensions of the form (4) generates an integer seuence (b n ). Integrality of the seuences defined by (4) is a conseuence of a version of the Laurent henomenon roved in [13] for cluster sueralgebras. 1.3. Weighted uivers. Our main tool is what we call a weighted uiver. This is a usual uiver Q (without 1- or 2-cycles), together with a function w : Q 0 Z on the set of vertices. Quiver mutations for such weight functions are defined as follows. Label the vertices by ositive integers 1,..., n, the mutation µ k at kth vertex sends w to the new function µ k (w) defined by: µ k (w)(i) = w(i) + [b ki ] + w(k), i k, µ k (w)(k) = w(k), where [b ki ] + is the number of arrows from the vertex k to the vertex i, and if the vertices are oriented from i to k, then [b ki ] + = 0. The exchange relations are also modified. Let us mention that the mutation rule of the weight function that we use (but not the exchange relations), have already been introduced by several authors; see formula (2.3) of [9] and [8],[2]. It would be interesting to investigate the relations of our work with these aers. We classify the eriod 1 uivers (in the sense of Fordy-Marsh [6]) that have a eriod 1 weight function.

DUAL NUMBERS, WEIGHTED QUIVERS, AND EXTENDED SOMOS AND GALE-ROBINSON SEQUENCES 3 2. Extensions of the Somos-4 seuence We start with the extensions of the Somos-4 seuence, briefly considered in [13]. The goal of this section is to show that the new seuences arising in this way have nice roerties. This section is based on a comuter rogram written by Michael Somos, to whom we are most grateful. It can be considered as a motivation for the rest of the aer. 2.1. Linearization. Consider first the recurrence (5) A n+4 A n = A n+3 A n+1 + A 2 n+2, where A n = a n + b n ε as in (1). The seuence (a n ) n N is then the Somos-4 seuence: a n = 1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898,... while the seuence (b n ) n N satisfies: b n+4 a n + b n a n+4 = a n+1 b n+3 + 2a n+2 b n+2 + a n+3 b n+1, which is the linearization of the Somos-4 recurrence. The sace of solutions of the linearized system is a four-dimensional vector sace. Every seuence (b n ) n N satisfying this recurrence is a linear combination of the seuences with one of the following initial conditions: (b 1, b 2, b 3, b 4 ) = (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), or (0, 0, 0, 1). Let us denote these seuences by (b 1 n) n N, (b 2 n) n N, (b 3 n) n N, (b 4 n) n N, resectively. We have the following roerties: (1) The four seuences (b 1 n) n N, (b 2 n) n N, (b 3 n) n N, (b 4 n) n N are integer. (2) The Somos-4 seuence is their sum, i.e., for every n. a n = b 1 n + b 2 n + b 3 n + b 4 n, Integrality of each of the four seuences (b i n) n N follows from the classic Laurent henomenon. Indeed, A n with n 5 is a Laurent olynomial in A 1, A 2, A 3, A 4. However, the denominators of this Laurent olynomial are monomials in a 1, a 2, a 3, a 4, while b 1, b 2, b 3, b 4 enter (linearly) into the numerators. Indeed, since ε 2 = 0, one has 1 a + bε = 1 a b a 2 ε. Proerty (2) follows from the fact that the seuences (b n ) n N satisfying the recurrence (5) form a vector sace, and the initial condition of the sum (b 1 n +b 2 n +b 3 n +b 4 n) n N is recisely (1, 1, 1, 1), i.e., that of (a n ) n N. Proerty (2) means that the seuences (b 1 n) n N, (b 2 n) n N, (b 3 n) n N, (b 4 n) n N rovide a canonical way to decomose the Somos-4 seuence with resect to the initial conditions. Note, however, that the seuences (b 1 n) n N and (b 2 n) n N are not ositive, the first values being: b 1 n = 1, 0, 0, 0, 2, 2, 10, 46, 103, 933, 4681, 27912, 375536,... b 2 n = 0, 1, 0, 0, 1, 2, 2, 1, 40, 140, 696, 265, 38478,... b 3 n = 0, 0, 1, 0, 2, 4, 5, 48, 94, 635, 4732, 18594, 299835,... b 4 n = 0, 0, 0, 1, 1, 3, 10, 22, 108, 472, 2174, 17792, 120536,... It would be interesting to understand the roerties of these seuences. Comuter calculations show that b 1 n < 0, for every n 100. The seuence (b 2 n) n N becomes ositive for n 15 (this checked for n 100). Conjecturally, the seuence (b 3 n) n N is ositive. The seuence (b 4 n) n N is ositive for n 25, however, b 4 n < 0, for 26 n 100.

4 VALENTIN OVSIENKO AND SERGE TABACHNIKOV 2.2. Two non-linear extensions. Consider now the extensions of the Somos-4 seuence, satisfying the recurrences: A n+4 A n = A n+3 A n+1 + A 2 n+2 (1 + wε) A n+4 A n = A n+3 A n+1 (1 + wε) + A 2 n+2, where w Z is an arbitrary (fixed) integer. The corresonding recurrences for (b n ) n N are nonlinear: (6) (7) b n+4 a n = a n+1 b n+3 + 2a n+2 b n+2 + a n+3 b n+1 b n a n+4 + w a 2 n+2, b n+4 a n = a n+1 b n+3 + 2a n+2 b n+2 + a n+3 b n+1 b n a n+4 + w a n+3 a n+1, resectively, where (a n ) n N is the initial Somos-4 seuence. It was roved in [13] that, for any choice of integral initial conditions, the seuence (b n ) n N satisfying (6) or (7) is integer. The choice of zero initial conditions: (8) (b 1, b 2, b 3, b 4 ) = (0, 0, 0, 0) is now the most natural one. Indeed, any solution (b n ) of each of the recurrences (6) and (7) is the sum of the solution with the initial condition (8) and a solution of the linear recurrence (5). In other words, the solutions of (6) and (7) form an affine subsace. Choosing w = 1, the seuences (b n ) n N defined by the recurrences (6) and (7) and zero initial conditions (8) start as follows: b n = 0, 0, 0, 0, 1, 2, 10, 48, 160, 1273, 7346, 51394, 645078, 5477318, 87284761... b n = 0, 0, 0, 0, 1, 3, 10, 59, 198, 1387, 9389, 57983, 752301, 6851887, 97297759... resectively. We conjecture the ositivity of these seuences, and we checked it for n 100. Furthermore, conjecturally, both of the above seuences grow faster than (a n ) n N. 3. Weighted uivers, mutations and exchange relations In this section, we introduce the notion of weighted uiver. We then describe the mutation rules of such objects, extending the usual mutation rules of uivers. We also describe the modified exchange relations for weighted uivers, and formulate the corresonding Laurent henomenon. The notion of weighted uiver is euivalent to the simlified version of cluster sueralgebra with two odd variables [13]. 3.1. Mutation rules. A uiver Q is an oriented finite grah with vertex set Q 0 and the set of arrows Q 1. Usually, the vertices of Q will be labeled by the letters {,..., x N }, where N = Q 0, considered as formal variables. In [3], Fomin and Zelevinsky defined the rules of mutation of a uiver, under the assumtion that Q has no 1-cycles and 2-cycles. The structure of Q can then be reresented as an N N- skew-symmetric matrix (b ij ), where b ij is the number of arrows between the vertices x i and x j. Note that the sign of b ij deends on the orientation: b ij > 0 if the arrows are oriented from x i to x j and negative otherwise. The mutation of the uiver µ k : Q Q at vertex x k is defined by the following three rules: for every ath (x i x k x j ) in Q, add an arrow (x i x j ); reverse all the arrows incident with x k ; remove all 2-cycles created by the first rule. Definition 3.1. We call a weighted uiver a uiver Q with a function w : Q 0 Z. The function w associates to every variable x i its weight, w i := w(x i ).

DUAL NUMBERS, WEIGHTED QUIVERS, AND EXTENDED SOMOS AND GALE-ROBINSON SEQUENCES 5 The mutation µ k (w) of the weight function w is erformed according to the following two rules: (1) for every arrow (x k x i ), change the value w i to w i + w k. In other words, the new weight function µ k (w) is defined by where for all i k; (2) reverse the sign of w k, i.e., µ k (w)(x i ) := w i + [b ki ] + w k, [b ki ] + = { bki, if b ki 0, 0, otherwise µ k (w)(x k ) := w k. 3.2. Exchange relations. Recall that the mutation µ k of the uiver Q relaces the variable x k by the new function x k defined by the formula: x k x k = x j + x i, x k x j x i x k where the roducts are taken over the set of arrows (x i x k ) Q 1 and (x k x j ) Q 1, resectively (with fixed k). The above formula is called the exchange relation. The Laurent henomenon, roved in [3], states that every (rational) function obtained by a series of mutations is actually a Laurent olynomial in the initial variables {,..., x N }. Given a weighted uiver (Q, w), we assume that the vertices are labeled by the variables {X 1,..., X N } written as dual numbers: X i = x i + y i ε, where x i and y i are the usual commuting variables. The exchange relations are defined as follows. Definition 3.2. The mutation µ k of (Q, w) relaces the variable X k by a new variable, X k, defined by the formula (9) X k X k = X j + (1 + w k ε) X i ; X k X j X i X k the other variables remain unchanged. This is a articular case of the exchange relations defined in [13]. 3.3. Laurent henomenon. The following version of Laurent henomenon is roved in [13] (this is the simlest case of Theorem 1). Theorem 3.3. For every weighted uiver (Q, w), all the functions X k, X k,..., obtained recurrently by any series of consecutive mutations, are Laurent olynomials in the initial coordinates {X 1,..., X N }. Remark 3.4. Note that Laurentness in {X 1,..., X N } means that the denominators are monomials in the variables {,..., x N }, while the variables {y 1,..., y N } enter (linearly) into the numerators.

6 VALENTIN OVSIENKO AND SERGE TABACHNIKOV 4. Period 1 weighted uivers Period 1 uivers were introduced and classified in [6]. These are uivers for which there exists a vertex such that the uiver rotates under the mutation at this vertex. More recisely, a eriod 1 uiver remains unchanged after the mutation comosed with the shift of the indices of the vertices i i 1. In this section, we answer the uestion which eriod 1 uivers have eriod 1 weight functions. A eriod 1 uiver euied with a eriod 1 weight function guarantees the integrality of seuence (b n ) n N defined by (4). 4.1. Examles. We start with simle examles. Examle 4.1. a) Consider the following weighted uiver with three vertices. After mutation at the uiver rotates, together with the weight function: w( ) = 1, w(x 2 ) = 0, w(x 3 ) = 1. µ 1 = We will say in such a situation, that the weight function has eriod 1. b) On the other hand, for the uiver with the inverted orientation: x 1 x 1 x 3 x 2 w(x 1) = 1, w(x 2 ) = 1, w(x 3 ) = 0. x 3 x 2 which is also of eriod 1, there is no weight function of eriod 1. Examle 4.2. Similarly, for the following uivers of eriod 1 with four vertices and the oosite orientations: a) x 4 and b) x 4 x 1 the weight function w( ) = 1, w(x 2 ) = 0, w(x 3 ) = 0, w(x 4 ) = 1, has eriod 1 in the first case, and there is no such function in the second case. Examle 4.3. Another interesting examle is the following Somos-4 uivers (cf. [6] and [10]): a) x 4 and b) x 4 Period 1 weight function exists for both choices of the orientation: resectively. w( ) = 1, w(x 2 ) = 0, w(x 3 ) = 0, w(x 4 ) = 1, w( ) = 1, w(x 2 ) = 1, w(x 3 ) = 1, w(x 4 ) = 1,

DUAL NUMBERS, WEIGHTED QUIVERS, AND EXTENDED SOMOS AND GALE-ROBINSON SEQUENCES 7 4.2. Period 1 weight functions: criterion of existence. Period 1 uivers were classified in [6]. The rimitive uiver P (t) N, where 1 t N 2, is a uiver with n vertices and n arrows, such that every vertex x i is joined with the vertex x i+t ( mod N) where the indices are always taken in the set {1,..., N}. The arrow is oriented from the vertex with the greater label to the vertex with the smaller label. One then has for P (t) N : b ij = 1, j i = t, 1, i j = t, 0, else. For instance, the uiver considered in Examles 4.1 b) and 4.2 b) are the uivers P (1) 3 and P (1) 4, resectively. Given a uiver Q, the oosite uiver Q is obtained by reversing the orientation. For examle, the uivers in Examles 4.1 a) and 4.2 a) are the uivers P (1) 3 and P (1) 4, resectively. More generally, if c Z, the uiver c Q is obtained by multilying the number of arrows between every two vertices, x i and x j by c. Finally, a sum of two uivers is obtained by suerosition of their arrows. It was roved in [6], that every eriod 1 uiver can be obtained as a linear combination of socalled rimitive uivers and a correction term. More recisely, let c 1,..., c r be arbitrary integers, where r = [ ] N 2. A eriod 1 uiver is of the form: (10) Q = c 1 P (1) N + + c rp (r) N + Q, where Q is a uiver with the vertices x 2,..., x N. Since we will only consider the mutation at, this correcting term Q will not change the exchange relations. We thus omit the exlicit form of Q ; see [6] and [10]. { c, c 0 Let us use the notation [c] = 0, otherwise. Theorem 4.4. Given a eriod 1 uiver Q, there exists a eriod 1 weight function on Q if and only if (11) [c 1 ] + + [c r ] = 1, if N is odd 2[c 1 ] + + 2[c r 1 ] + [c r ] = 2, if N is even. The eriod 1 weight function is uniue u to an integer multile. Proof. Assume that a eriod 1 uiver Q admits a eriod 1 weight function w. By definition 3.1, the mutation at the first vertex, µ 1, transforms the weight function as follows: w 1 w 1, w i w i + [b 1i ] + w 1, for all 1 i N. Since w is of eriod 1, this imlies the following system of linear euations: w n = w 1, w 1 = w 2 + [b 12 ] + w 1, w 2 = w 3 + [b 13 ] + w 1, w N 1 = w n + [b 1N ] + w 1, that has (a uniue) solution if and only if the following condition is satisfied: [b 12 ] + + + [b 1N ] + = 2.

8 VALENTIN OVSIENKO AND SERGE TABACHNIKOV Finally, from (10), one has [b 1i ] + = [c i 1 ], if i r and [b 1i ] + = [c N i+1 ], if i r. The above necessary and sufficient condition for the existence of the function w then coincides with (11). 5. Alications to integer seuences We aly the above constructions to integer seuences. 5.1. The general method. Given a weighted uiver (Q, w) of eriod 1, by Theorem 3.3, erforming an infinite series of consecutive mutations: µ 1, µ 2,..., one obtains a seuence of Laurent olynomials (X n ) n N in the initial variables {X 1,..., X N }. This seuence satisfies the recurrence: X n+n X n = X [b1i]+ n+i (1 + w 1 ε) + X [b1i] n+i. 1 i N 1 1 i N 1 Recall that X i = x i +y i ε. Choosing the initial conditions (,..., x N ) := (1,..., 1) and arbitrary integers (y 1,..., y N ) := (b 1,..., b N ), one obtains a seuence (A n ) n N, where A n = a n + b n ε. The constructed integer seuence (b n ) n N is the desired extension of (a n ) n N. Let us give further examles. 5.2. Seuences of order 2. We illustrate the idea of substitution of dual numbers into recurrences on a very simle classic examle. Consider the classic Fibonacci numbers (F n ) = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... and let us slit (F n ) into two subseuences: a n := F 2n 1, ã n := F 2n. Both of them satisfy uadratic recurrences that differ by a sign: (12) a n+2 a n = a 2 n+1 + 1, ã n+2 ã n = ã 2 n+1 1. The above recurrences are known as Cassini s identity. The initial conditions for these seuences are: (a 0, a 1 ) = (1, 1) and (ã 0, ã 1 ) = (0, 1). We will consider the seuences of dual numbers: with the recurrence relations generalizing (12). A n := a n + b n ε, Ã n := ã n + b n ε, Linearization: dual Fibonacci and Lucas numbers. Suose that (A n ) and (Ãn) satisfy the similar recurrences: (13) A n+2 A n = A 2 n+1 + 1, Ã n+2 Ã n = Ã2 n+1 1. Euivalently, the seuences (a n ) and (ã n ) are as above, and (b n ) and ( b n ) are defined by: (14) b n+2 a n = 2b n+1 a n+1 b n a n+2, bn+2 ã n = 2 b n+1 ã n+1 b n ã n+2. The seuence (b n ) is integer for an arbitrary choice of integral initial conditions (b 0, b 1 ) = (, ). It turns out that the classic Lucas numbers naturally aear in the dual Fibonacci seuences. Proosition 5.1. The seuences of odd (res. even) Lucas numbers: satisfy the recurrence (13). b n = L 2n 1, bn = L 2n, Proof. Using the exlicit formulas for the Fibonacci and Lucas numbers F n = ϕn ( ϕ) n 5, L n = ϕ n + ( ϕ) n, where ϕ is the golden ratio, one checks directly that the recurrences (14) are satisfied.

DUAL NUMBERS, WEIGHTED QUIVERS, AND EXTENDED SOMOS AND GALE-ROBINSON SEQUENCES 9 The Lucas solutions to (13) start as follows: n 0 1 2 3 4 5 6 7 a n 1 1 2 5 13 34 89 233 b n 1 1 4 11 29 76 199 521 n 0 1 2 3 4 5 6 7 ã n 0 1 3 8 21 55 144 377 bn 2 3 7 18 47 123 322 843 Furthermore, one has a 2-arameter family of solutions to the recurrence (13): n 0 1 2 3 4 5 6 7 a n 1 1 2 5 13 34 89 233 b n 2 + 2 8 + 3 27 + 2 86 10 265 66 798 277 The situation is more comlicated for the seuence ( b n ) n Z. Arbitrary initial conditions ( b 0, b 1 ) do not lead to an integer seuence. But one obtains a two-arameter family of integer seuences by choosing the initial conditions ( b 1, b 2 ) = (3, ) with arbitrary and. n 1 2 3 4 5 6 7 ã n 1 3 8 21 55 144 377 bn 3 6 24 25 128 90 507 300 1778 954 5835 Note that the seuence of coefficients of is A001871. Liming Fibonacci seuence. The classic odd Fibonacci seuence a n = F 2n 1 satisfies the first recurrence in (12). It can be generated by consecutive mutations µ 0, µ 1, µ 0, µ 1,... of the uiver with two vertices and two arrows: x 0, called the 2-Kronecker uiver. Clearly, there is no eriod 1 weight function w, but the function w 1 has eriod 4. The seuence of consecutive mutations at vertices, x 2,,... then leads to the recurrence ) (15) A n+2 A n = A 2 n+1 (1 + ( 1) (n+1)(n+2) 2 ε + 1. More recisely, (b n ) n Z satisfies b n+2 a n + b n a n+2 = 2b n+1 a n+1 + ( 1) (n+1)(n+2) 2 a 2 n+1. Let us consider the initial conditions a 0 = a 1 = 1 and b 0 = b 1 = 0. It turns out that the seuence (b n ) n N also consists of Fibonacci numbers, but this time with even indices, and taken in a surrising order: n 0 1 2 3 4 5 6 7 a n 1 1 2 5 13 34 89 233 b n 0 0 1 8 21 21 55 377 More recisely, b n = { F2n, n 0, 3 mod 4, F 2n 2, n 1, 2 mod 4.

10 VALENTIN OVSIENKO AND SERGE TABACHNIKOV 5.3. Seuences of order 3. Consider the seuence A005246 satisfying the recurrence 1 a n+3 a n = a n+2 a n+1 + 1, and starting as follows: a n = 1, 1, 1, 2, 3, 7, 11, 26, 41, 97, 153, 362, 571, 1351,... This seuence can be generated by the eriod 1 uivers a) and b) x 1 x 3 x 2 already considered in Examle 4.1. By Theorem 4.4, the first uiver has a eriod 1 weight function, but not the second one. Therefore, the seuence (b n ) n N defined by the recurrence A n+3 A n = A n+2 A n+1 + 1 + w ε and arbitrary integer initial conditions is integer. Our results do not give any information about the seuence (b n ) n N satisfying A n+3 A n = A n+2 A n+1 (1 + ε) + 1, but numerical exeriments show that it is not integer. However, consider again the uiver b). The weight function w( ) = w(x 2 ) = w(x 3 ) = 1 is of eriod 6. Indeed, after three consecutive mutations µ 3 µ 2 µ 1, the function w changes its sign and becomes w(x 1) = w(x 2) = w(x 3) = 1, while the uiver remains unchanged. Therefore, the recurrence ( ) (16) A n+3 A n = A n+2 A n+1 1 + ( 1) (n+1)(n+2)(n+3) 6 ε + 1 defines integer seuences (b n ) n N. Note that the exonent is chosen to obtain the sign seuence +, +, +,,,, +, +, +,... Written more exlicitly, (b n ) n N satisfies the non-linear recurrence b n+3 a n = b n+2 a n+1 + b n+1 a n+2 b n a n+3 + ( 1) (n+1)(n+2)(n+3) 6 a n+2 a n+1. For examle, zero initial conditions lead to the following seuence b n = 0, 0, 0, 1, 3, 15, 17, 43, 2, 112, 84,... 5.4. Non-homogeneous Somos-4 seuence. Let and be ositive integers, and consider the seuence (a n ) n N defined by the recurrence a n+4 a n = a n+3 a n+1 + a n+2 and the initial conditions a 0 = a 1 = a 2 = a 3 = 1. This seuence was considered in [7]; see also [6]. Corollary 5.2. (i) The seuence (b n ) n N, defined by the recurrence (17) A n+4 A n = A n+3 A n+1 + A n+2 (1 + ε) with arbitrary integer initial conditions (b 1, b 2, b 3, b 4 ), is integer. (ii) The seuence (b n ) n N, defined by the recurrence (18) A n+4 A n = A n+3 A n+1 (1 + ε) + A n+2 with arbitrary integer initial conditions (b 1, b 2, b 3, b 4 ), is integer. 1 Note that, unlike the Somos seuences, this seuence also satisfies a linear recurrence: an+4 = 4a n+2 a n.

DUAL NUMBERS, WEIGHTED QUIVERS, AND EXTENDED SOMOS AND GALE-ROBINSON SEQUENCES 11 Proof. Following [6], consider the uivers (that differ only by orientation): (19) x 4 and x 4 (+1) (+1) where the labels, and ( + 1) stand for the number of arrows. Each of them rotates under the series of consecutive mutations µ 1, µ 2, µ 3,... For instance, x 4 = µ 1 x4 x 1 (+1) (+1) This is straightforward from the definition of uiver mutations (and similarly for the twin uiver). By Theorem 4.4, the first of the uivers (19) has a weight function of eriod 1, if (and only if) = 1, while the second uiver has a weight function of eriod 1, if (and only if) = 2. Note that our results do not imly the converse statement. However, we conjecture that the seuence (b n ) n N, defined by the recurrence A n+4 A n = A n+3 A n+1 + A n+2 (1 + ε), is integer if and only if = 1 (and similarly for the second case). This conjecture is confirmed by the following examles. Examle 5.3. Let us now consider the seuence (A n ) n N satisfying the recurrence A n+4 A n = A n+3 A n+1 (1 + ε) + A n+2, with initial conditions: b 0 = b 1 = b 2 = b 3 = 0, and take 2. Although Theorem 4.4 does not imly non-integrality of (b n ) n N, this seuence is not integer in all the examles we considered. a) If = 0, then the seuence starts as follows: n 0 1 2 3 4 5 6 7 8 9 a n 1 1 1 1 2 3 4 9 14 19 307 b n 0 0 0 0 1 3 6 24 56 3 b) If = 1, then the seuence stos to be integer one ste earlier: n 0 1 2 3 4 5 6 7 8 a n 1 1 1 1 2 3 5 13 22 159 b n 0 0 0 0 1 3 7 32 2 c) If = 3, then, the seuence is: n 0 1 2 3 4 5 6 7 8 a n 1 1 1 1 2 3 11 49 739 6539 b n 0 0 0 0 1 3 18 150 2 This and many other exerimental comutations illustrate a sohisticated and fragile nature of the Laurent henomenon of Theorem 3.3. It seems to occur only when there is a weighted function with eriod 1.

12 VALENTIN OVSIENKO AND SERGE TABACHNIKOV 5.5. Conclusion and an oen roblem. The roerties of the constructed integer seuences remain unexlored. In many cases, we cannot rove their ositivity (although this was checked numerically for the most interesting examles), and their asymtotics are unknown. A very interesting roerty of the Somos-tye seuences is their relation to discrete integrable systems; see [5] and references therein. (The roerties of integrality and integrability are related not only honetically!) It will be interesting to investigate integrability of discrete dynamical systems related to the seuences constructed in this aer. For examle, is the following ma on R 8 x 2 x 3 x 4 y 1 y 2 y 3 y 4 x 2 x 3 x 4 ( ) x4 x 2 + x 2 3 /x1 y 2 y 3 y 4 ( ) ( ) x2 y 4 + 2x 3 y 3 + x 4 y 2 + x 2 3 /x1 y 1 x4 x 2 + x 2 3 /x 2 1 that arises from the extended Somos-4 recurrence (see Section 2) comletely integrable? Acknowledgements. This aer was initiated by discussions with Michael Somos; we are indebted to him for many fruitful comments and a comuter rogram. We are grateful to Gregg Musiker and Michael Shairo for enlightening discussions. This aer was comleted when the first author was a Shairo visiting rofessor at Pennsylvania State University, V.O. is grateful to Penn State for its hositality. S.T. was artially suorted by the NSF Grant DMS-1510055. References [1] M.Bousuet-Mélou, J.Pro, J.West. Perfect matchings for the three-term Gale-Robinson seuences, Electron. J. Combin. 16:1 (2009), 1 37. [2] J.A. Cruz Morales, S. Galkin, Uer bounds for mutations of otentials, SIGMA Symmetry Integrability Geom. Methods Al. 9 (2013), Paer 005, 13. [3] S. Fomin, A. Zelevinsky, Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15 (2002), 497 529. [4] S. Fomin, A. Zelevinsky, The Laurent henomenon. Adv. in Al. Math. 28 (2002), 119 144. [5] A. Fordy, A. Hone, Discrete integrable systems and Poisson algebras from cluster mas, Comm. Math. Phys. 325 (2014), 527 584. [6] A. Fordy, R. Marsh, Cluster mutation-eriodic uivers and associated Laurent seuences, J. Algebraic Combin. 34 (2011), 19 66. [7] D.Gale, The strange and surrising saga of the Somos seuences. Math. Intell. 13, (1991) 40 42. [8] S. Galkin, A. Usnich, Mutations of otentials, Prerint IPMU 10-0100, 2010. [9] M. Gross, P. Hacking, S. Keel, Birational geometry of cluster algebras, Algebr. Geom. 2 (2015), 137 175. [10] R. Marsh, Lecture notes on cluster algebras, Zurich Lectures in Advanced Mathematics. Euroean Mathematical Society (EMS), Zürich, 2013. [11] S. Morier-Genoud, Coxeter s frieze atterns at the crossroads of algebra, geometry and combinatorics, Bull. Lond. Math. Soc. 47 (2015), 895 938. [12] S. Morier-Genoud, V. Ovsienko, S. Tabachnikov. Introducing suersymmetric frieze atterns and linear difference oerators, Math. Z. 281 (2015), 1061 1087. [13] V. Ovsienko, A ste towards cluster sueralgebras, arxiv:1503.01894. [14] D. Seyer, Perfect matchings and the octahedron recurrence, J. Algebraic Combin. 25 (2007), 309 348. [15] E. Study, Geometrie der Dynamen. Leizig, 1903. Valentin Ovsienko, CNRS, Laboratoire de Mathématiues U.F.R. Sciences Exactes et Naturelles Moulin de la Housse - BP 1039 51687 REIMS cedex 2, France

DUAL NUMBERS, WEIGHTED QUIVERS, AND EXTENDED SOMOS AND GALE-ROBINSON SEQUENCES 13 Serge Tabachnikov, Pennsylvania State University, Deartment of Mathematics, University Park, PA 16802, USA, E-mail address: valentin.ovsienko@univ-reims.fr tabachni@math.su.edu