An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si.Mat. (N.S.) Tomul LXVI, 2020, f. 1

Padovan-like sequences and generalized Pascal’s triangles

Giuseppina Anatriello Giovanni Vincenzi ·

Abstract The Padovan sequence and the Plastic number are mathematical objects of central interest among architects, and from this point of view, their importance is com- parable to that of the Fibonacci sequence and the Golden mean. Padovan-like sequences, such as Perrin sequence or the Cordonnier sequence, also have mathematical applications. In this article, we will characterize a large family of Padovan-like sequences in terms of ‘Padovan diagonals’ of generalized Pascal’s triangles. Keywords Padovan-like sequences Plastic number Combinatorial identities Kepler limit Generalized Pascal’s triangle · Computational methods· · · · Mathematics Subject Classification (2010) 05A19 03D20 00A67 11Y55 · · ·

1 Introduction

The Fibonacci sequence, which we will denote by (Fn), is defined by

F0 = 0,F1 = 1 Fn = Fn 1 + Fn 2, n > 1, (1.1) − − and it has been of wide interest since his first appearance in the book Liber Abaci published in 1202. Four centuries later, Kepler showed (see the book “Harmonices Mundi”, 1619, p. 273, see [20], [21]) that: F lim n+1 = Φ (1.2) n Fn where Φ denotes the highly celebrated Golden mean, which appears in sev- eral human fields, especially in Art and Architecture, even if sometimes not always appropriately (see e.g. [14], [24] and references therein). Among the most relevant generalizations of the Fibonacci sequence, there is the Tribonacci sequence (Tn). It is a third order recursive sequence defined by

T0 = 0,T1 = T2 = 1, and Tn = Tn 1 + Tn 2 + Tn 3, n > 1. (1.3) − − −

25 2 Giuseppina Anatriello, Giovanni Vincenzi

Both Fibonacci and Tribonacci sequences, and their generalizations can serve as a model of recursive phenomena in Botany (see e.g. [30], [12]), Chemistry (see e.g. [29]), Medicine (see e.g. [28]). Many authors have investigated recursive sequences (e.g. [4], [3], [7], [18], [23]) from different points of view, in particular for applications in many problems of computer science ([17], [37]), Physics and Engineering (see e.g. [4] [16] [13] [9] and references therein). Another third order homogeneous linear recursive sequence that certainly should be mentioned in this fashion is the Padovan sequence (see e.g. [1], [8], [22], [24], [25], [26], [27], [32], [35]):

1, 1, 1, 2, 2, 3, 4, 5, 7, 9,...

It is a sequence of integer numbers, usually denoted by (Pn) and defined by

P1 = P2 = P3 = 1 Pn = Pn 2 + Pn 3, n > 3. (1.4) − −

Clearly, Pn can be defined for every integer n, and in particular P0 = P3 P1 = 0. As− a particular case of a general theory (see [2], [10]), it can be shown that lim Pn/Pn 1 is the positive real solution of the characteristic (Cauchy) n − 3 polynomial x x 1 of the sequence (Pn). This number, whose value is about 1.32471795724474− − ... is called the Plastic number, and it is denoted by Ψ:

P lim n+1 = Ψ. (1.5) n Pn The Plastic number has many surprising properties, which remind to the Golden mean. For example (see [27], p. 370) it satisfies the following equa- tions: Ψ 3 = Ψ +1,Ψ 5 = Ψ 2 +Ψ +1 = Ψ 4 +1,Ψ 8 = Ψ 4 +Ψ 3 +Ψ 2 +Ψ +1.

Both the notions of Padovan sequence and Plastic number were being introduced by H. van der Laan (Leiden, 29 December 1904 - Mamelis, 19 August 1991), a Dutch Benedictine monk who was one of the most relevant architects of the last century ( see pp. 181-193 in [25] or [26] pp. 407-419). The main aim of van der Laan was to find the true foundations of ar- chitecture, based on mathematics, spirituality and harmony. For these fi- nalizations, he founded the cultural circle of studies “Bouwkundige Studie Kring”. The Padovan sequence and the Plastic number are mathematical objects of central interest among architects, and from this point of view, their importance is comparable to that of Fibonacci sequence and Golden mean: actually, the Plastic number is considered to be a three-dimensional expression of the golden ratio. Richard Padovan wrote (see [26]):

26 Padovan-like sequences 3

The Plastic number discovered by Dom Hans van der Laan differs from all previous systems of architectural proportions in several fun- damental ways. Its derivation from a cubic equation (rather than a quadratic one such as that which defines the golden section) is a re- sponse to the three-dimensionality of our world. Its basic ratios, ap- proximately 3 : 4 and 1 : 7, are determined by the lower and up- per limits of our normal ability to perceive differences of size among three-dimensional objects. Proportion plays a crucial role in generating architectonic space, which comes into being through the proportional relations of the solid forms that delimit it. Architectonic space might therefore be described as a proportion between proportions.

It is known (see Figure 1) that both Fibonacci and Padovan sequences share the property to be found in the Pascal’s triangle, by means of ‘diago- nals’.

Fig. 1 The Fibonacci and the Padovan sequences detected from Fibonacci and the Padovan diagonals in the Pascal’s triangle. Note that the vertex of the triangle on the right must be 0 if we want P0 = 0.

In the following, we will refer to such diagonals respectively as Fibonacci diagonals and Padovan diagonals. Along this paper, given a whichever configuration of number disposed in triangular way, say T , for any i 0 and 0 j i we will denote by Q(i, j) the (i, j) entry of T , that is the≥ element that≤ ≤ lies on the j site (counting from the left to the right, starting from 0) of the i-th row of T . For example if T is the above Pascal triangle’s we have Q(3, 0) = 1, Q(3, 1) = 3... It can be noted that the inclination of Fibonacci diagonals is slightly greater than the inclination of Padovan diagonals. Precisely, if an element Q(i, j) lies on a Fibonacci diagonal, then the element above on the same diagonal shifts of one position, so it is Q(i 1, j+1); while if Q(i, j) lies on a Padovan diagonal, then the element above− on the same diagonal shifts of two positions, so it is Q(i 1, j + 2). There− are many authors who have studied the connections between special families of recursive sequences and generalized Pascal’s triangles (for a rich compendium see [5]). Recall that given two integers k1 and k2 (indeed, in the whole paper one should consider a whichever pair of elements k1 and k2 of a commutative

27 4 Giuseppina Anatriello, Giovanni Vincenzi ring), the Fibonacci-Like sequence (Hn) whose seeds are H1 = k1 and H2 = k2, is the sequence defined by the recursive rule

Hn = Hn 1 + Hn 2, n > 2. − − ∀

Similarly (following [32] and [3]), given three elements k1, k2 and k3 of a commutative ring, the Padovan-Like sequence (Hn) whose seeds are H1 = k1, H2 = k2 and H3 = k3, is the sequence defined by the recursive rule

Hn = Hn 2 + Hn 3, n > 3. − − ∀ Relevant examples of Padovan-like sequence are the Perrin sequence and the Cordonnier sequence (see [32]). Note that some other uses this definition for something of a different (see [15]). The first connection between Pascal’s triangle and Fibonacci sequence was established by Lucas (see for instance [6], p.112):

n n 1 n i Fn+1 = + − + ... − + ... i : n i i 0. (1.6) 0 1 i ∀ − ≥ ≥       This identity shows that Fibonacci diagonals determine exactly the Fi- bonacci sequence (see Figure 1 on the left).

Fig. 2 The sequence (Dn) associated with the Generalized Pascal’ s triangle T (k1, k2).

This result admits many natural extensions ( see [19], [31] [33], [36], just some of the more recent ones), in particular Siani and Vincenzi [36], showed that every Fibonacci-like sequence can be obtained as the sequence of di- agonals associated to a generalized Pascal’s triangle (see Figure 2); and conversely every sequence associated to a generalized Pascal’s triangle is a Fibonacci-like sequence (Theorem 2.1, Equation 3 and Equation 4 in [36]). Recall that starting from k1 on the left edge, k2 on the right one and the vertex being k2 (see Figure 2 and [36] for other details), the generalized Pascal’s triangle T (k1, k2) is the triangle consisting of numbers, constructed in the usual way (each term is the sum of the two above). In particular, T (1, 1) is the usual Pascal’ s triangle.

28 Padovan-like sequences 5

In this paper we will extend the above results. Taking [36] as model, in section 2, we will define in a natural way the ‘sequences (∆n) associated with the Padovan diagonals of generalized Pascal’s triangles’ (see Figure 3),

Fig. 3 Padovan diagonals in the generalized Pascal’s triangle T 0(k1, k2).

and in Section 3 we will show a characterization of a large family of Padovan-like sequences in terms of ‘Padovan diagonals’ of generalized Pas- cal’s triangles.

2 Padovan diagonals and related properties

In this section, we establish some notations and preliminary properties of Padovan diagonals which will be useful in the following. If m is a positive integer, we will denote by ∆m indifferently both the m- th Padovan diagonal - which consists of the elements which lie on it - and the sum of such elements. The following remark gives the exact numbers of elements which lie on a Padovan diagonal. Recall that for every integers a, a a and b, the symbol [ b ] means the integer part of b . Remark 2.1 Note that denoting by Q(i, j) the element lying on the j-th position of the i-th row of a Pascal’s triangle (see introduction for details), we have: – if m = 2n, than

∆ = Q(n, 2 0+1),Q(n 1, 2+1),...,Q(n i, 2i+1), i such that n i 2i+1 . m { · − − ∀ − ≥ } Note that in describing ∆m, we must consider 0, and all the positive n 1 indexes i such that i −3 . Thus, in this case the number of the elements ≤ n 1 m 2 which lie on ∆m is 1 + [ −3 ] = 1 + [ 6− ]; – if m = 2n 1, than − ∆ = Q(n, 2 0),Q(n 1, 2 1),...,Q(n i, 2i), i such that n i 2i . m { · − · − ∀ − ≥ }

29 6 Giuseppina Anatriello, Giovanni Vincenzi

Similarly, in describing ∆m, we must consider 0, and all the positive in- dexes i such that i n/3. It follows in this case that the number of the ≤ n m+1 elements which lie on ∆m is: 1 + [ 3 ] = 1 + [ 6 ].

a Recall that for every pairs of integers a and b the binomial coefficient b is defined by  

a! a b!(a b)! if 0 b a = − ≤ ≤ . (2.1) b 0 if a < b or b < 0   ( By Remark 2.1, the sum of the elements lying of the m-th Padovan diag- onal of the classical Pascal’s triangle is: n n 1 n i + − + ... + − + ..., if m = 2n, 1 2 1 + 1 2i + 1    ·    [(n 1)/3] − (here the sums| stops when n {zi < 2i + 1); and } − n n 1 n i + − + ... − + ..., if m = 2n 1 0 2 2 i −      ·  [n/3] (here the sums stops| when n i{z < 2i). } It turns out that the sum− of the elements lying on the m-th Padovan diagonal of the classical Pascal’s triangle (Figure 1 on the right) is described by the following relations:

[(n 1)/3] [n/3] − n i n i S = − if m = 2n, and S = − if m = 2n 1. m 2i+1 m 2i − Xi=0   Xi=0   n n Note that n 3 < 2 3 + 1, therefore, taking into account the Definition − n n 3 (2.1), we have n− = 0, so that the above relations may be written as: 2 3 +1
[n/3] [n/3] n i n i S = − if m = 2n, and S = − if m = 2n 1. m 2i+1 m 2i − Xi=0   Xi=0   The above sums describe exactly the terms of the Padovan sequence, as it is Sm = Pm, m N. This follows from the well-known ‘Combinatorial Identity for Padovan∀ ∈ sequence’ (see Figure 1 on the right), which can be considered as the corresponding of Lucas Identity (see Equation 1.6 and Figure 1 on the left) for the Padovan sequence:

30 Padovan-like sequences 7

Theorem 2.1 (Combinatorial Identity for Padovan sequence) Let (Pm) be the Padovan sequence, then:

[n/3] [n/3] n i n i m>0,P = − if m=2n 1 and P = − if m=2n. ∀ m 2i − m 2i + 1 Xi=0   Xi=0  

Now let k1 and k2 be two integers, and consider the ‘Pascal’s triangle’ T 0(k1, k2) obtained from T(k1, k2), just replacing the vertex k2 with 0 (look at Figure 3). Note that, in order to have the element P0 of the Padovan sequence to be equal 0 as the 0-diagonal of the Pascal’s triangle. We have already done this modification in Figure 1 on the right. We will usually denote the Padovan diagonals with dashed orange lines (see Figure 1 on the right and Figure 3 for Padovan diagonals respectively referred to the Pascal’s triangles T 0(1, 1) and T 0(k1, k2)). The sequence (∆m) will be called the sequence associated with T 0(k1, k2). Looking at the Figure 3 it turns out that: ∆0 = 0, ∆1 = k1, ∆2 = k2, ∆3 = k1, ∆4 = k1 + k2, ∆5 = k1 + k2, ∆6 = 2k1 + k2, ∆7 = 2k1 + 2k2, ∆8 = 3k1 + 2k2. We will see in the next section that (∆m) is a Padovan-like sequence. We conclude this section noting that: Remark 2.2 The Kepler limit of every Padovan-like sequence whose ini- tial condition are real numbers is always the Plastic number, Ψ (see [2] Section 2.2 and 2.3). In particular the Kepler limit of Perrin sequence and Cordonnier sequence is Ψ. Note that this fails to be true for arbitrary triples of complex numbers (see [10], examples in Section 4, and [11]).

3 The main result

Clearly, the Padovan sequence can be defined also for negative index. In particular, we have:

...P 5 = 1,P 4 =1,P 3 = 0,P 2 = 0,P 1 = 1,P0 = 0,P1 = 1,P2 = 1,... − − − − − − The following result gives the closed formula for a Padovan-like sequence. Lemma 3.1 Let (H ) be a Padovan-like sequence. Then, m > 0 m ∀ Hm = Pm 5H1 + Pm 3H2 + Pm 4H3, where (Pm) is the Padovan sequence. − − − Proof. Clearly, the statement it is true if m = 1, 2, 3, 4. Let m > 4, and by induction suppose that Hm 2 = Pm 7H1 + Pm 5H2 + Pm 6H3, and − − − −

31 8 Giuseppina Anatriello, Giovanni Vincenzi

Hm 3 = Pm 8H1 + Pm 6H2 + Pm 7H3. It follows that − − − − Hm = (Pm 7 + Pm 8)H1 + (Pm 5 + Pm 6)H2 + (Pm 6 + Pm 7)H3, that is − − − − − − Hm = Pm 5H1 + Pm 3H2 + Pm 4H3. − − − We are now in a position to prove the main Theorem, which gives a con-ut nection between sequences associated with the generalized Pascal’s triangles and Padovan-like sequences. From now on by Q(i, j) we will denote the el- ement lying on the j-th position of the i-th row of a ‘generalized’ Pascal’s triangle T 0(k1, k2).

Theorem 3.2 Let k1 and k2 be complex numbers. Let (∆m) be the sequence associated with the generalized Pascal’s triangle T 0(k1, k2) and let (Hn) be the Padovan-like sequence of seeds H1 = 0,H2 = k1 and H3 = k2. Then

Hm+1 = ∆m = Pm 2k1 + Pm 3k2, m 0, − − ∀ ≥ where (Pm) is the Padovan sequence. In particular (∆m) is a Padovan like- sequence.

Proof. Put Q(0, 0) = 0 the vertex of T 0(k1, k2), and for any i 0 and 0 j i put Q(i, j) the (i, j) entry of T (k , k ). By construction ≥Q(i, j) is ≤ ≤ 0 1 2 the sum of the two elements that lie above it in T 0(k1, k2): Q(i, j) = Q(i 1, j 1) + Q(i 1, j). − − − Using the Stiefel’s identity, we see that for any i 0 the following relation holds: ≥ i 1 i 1 Q(i, j) = k − + k − , 0 j i. (3.1) 1 j 2 j 1 ∀ ≤ ≤    −  Now we can compute the Padovan diagonals ∆m. We split into two cases: Let 0 < m = 2n 1, then the first term of the m-th diagonal, which lies − on the left side of the triangle on the n-th line is Q(n, 0) = k1. The second element of ∆m is Q(n 1, 2), which lies on the n 1-th row of the triangle and its position shifts− of two places on the right.− So on, proceeding in this way (see Figure 3), it turns out that: ∆2n 1 = Q(n, 0) + Q(n 1, 2) + ... + Q(n i, 2i) + ..., where− the sum stops when−n i < 2i + 1. − So that by Equation (3.1),− we have n 1 n 2 n 3 n i 1 ∆2n 1 = k1[ − + − + − + ... + − − + ...] − 0 2 2 2 2i      ·    [n/3] n 1 n 2 n 3 n i 1 + k [ − + |− + − +{z... + − − + ...} ]. 2 2 0 1 2 1 1 2 2 1 2i 1  · −   · −   · −   −  [n/3] n n/3 1 If [n/3] = n/3 the binomial − − = 0; moreover, if we put h = (i 1) | 2n/3 {z } − we can rewrite the above relation as follows:

32 Padovan-like sequences 9

n 1 n 2 n 3 n i 1 ∆2n 1 = k1[ − + − + − + ... + − − + ...] − 0 2 2 2 2i      ·    [(n 1)/3] − (n 2) 0 (n 2) 1 (n 2) h + k [0 + − − +| − − + ... +{z − − + ...]. } 2 2 0 + 1 2 1 + 1 2h + 1  ·   ·    [(n 2)/3] − An application| of Theorem 2.1 yields{z }

∆2n 1 = k1P2(n 1) 1 + k2P2(n 2). (3.2) − − − − Now let 0 < m = 2n, then the first term of the m diagonal is (n 1)k +k − 1 2 which lies close to the left side of the triangle T 0(k1, k2) on the n-th line (see for example the red numbers in the Figure 3). As in the case above, the second element of ∆m lies on the n 1-th line and its position shifts of two places to the right, and so on the third...− It turns out that: ∆ = Q(n, 1) + Q(n 1, 1 + 2) + ... + Q(n i, 1 + 2i) + .... 2n − − So that by Equation (3.1), we have ∆2n n 1 n 2 n i 1 = k [ − + − + ... + − − ...] 1 1 1 + 2 1 + 2i       1+[(n 1)/3] − n 1 n 2 n i 1 + k [ | − + − + ...{z + − − + ...}], 2 0 2 2i       1+[(n 1)/3] and by Theorem 2.1 we have− | {z }

∆2n = k1P2(n 1) + k2P2(n 1) 1. (3.3) − − −

Consider now the Padovan-like sequence whose seeds are H1 = 0,H2 = k1,H3 = k2 (and whose low is Hm = Hm 2 + Hm 3). By Lemma 3.1, and by Equations (3.2)− and (3.3)− we have: Hm+1 = Pm 2k1 + Pm 3k2 = ∆m. The proof is complete. − − 

4 Conclusions

The idea to discover numerical harmony inside geometric configurations is one of the main goals for mathematicians. In our context, as a general problem, one could ask which recursive sequences can be described through geometrical objects. We have seen that recursive sequences may have applications in counting problems referred to graphs, and perhaps also should be mentioned that

33 10 Giuseppina Anatriello, Giovanni Vincenzi recursive sequences are useful in GPS systems (for localizations). Here we may aspect that other Padovan-like sequences may be useful both in archi- tecture and design, especially in digital process and design [34], than in a large spectrum of problems in mathematics (see [32]).

Acknowledgements This work was supported by the National Group of Algebraic and Geometric Structures and their Applications (GNSAGA–INDAM), Italy.

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Received: 23.I.2019 / Revised: 01.II.2019 / Accepted: 04.II.2019

Authors Giuseppina Anatriello, Dipartimento di Architettura, Universit`adi Napoli “Federico II”, Napoli, Italy, E-mail: [email protected]

Giovanni Vincenzi (Corresponding author), Dipartimento di Matematica, Universit`adi Salerno, Salerno, Italy, E-mail: [email protected]

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