Forum Geometricorum Volume 18 018) 47 55. FORUM GEOM ISSN 1534-1178 Integer Sequences and Circle Chains Inside a Circular Segment Giovanni Lucca Astract. We derive the conditions for inscriing, inside a circular segment, a chain of mutually tangent circles having the property that the ratio etween the largest circle radius and the radius of any other circle of the chain is an integer numer. 1. Introduction If we consider a straight line intersecting a circlec, the two areas ounded y the common chord and one of the two arcs are named circular segments. Inside each one of the circular segments it is possile to inscrie infinite chains of mutually tangent circles which are also tangent to the outer circle and to the common chord; a generic example of chain is shown in Figure 1. y G C major circle A O x B H Figure 1. Example of a circle chain inscried inside a circular segment In a previous paper [], we derived some geometrical properties and formulas related the circles forming the chain. Here, we follow a different approach, ased on the inversion technique, and investigate aout some connections that can e found etween the circle chains and certain integer sequences. Pulication Date: January 6, 018. Communicating Editor: Paul Yiu.
48 G. Lucca. Some definitions and useful expressions For the following, it is convenient to define the major circle in the chain see Figure 1) as the one having the largest radius and lael it y index 0; thus, we can sudivide a generic chain into two su-chains: an up chain starting from the major circle and converging to point G and a down chain starting from the major circle and converging to point H. Moreover, let a + ) e the diameter of the outer circle and the length of segmentob. By setting up a coordinate system with origin ato, we havea,b,g andh with coordinates a,0),,0),0, a) and0, a) respectively. In this paper, we want to investigate the conditions under which the ratios τ k, k 1,,..., etween the radius of the major circle and the generick-th circle are integers for oth the up and down su-chains. In other words, what are the conditions provided they exist) for the radii of the circles of the chain to e sumultiples of the major circle radius? From [], we report that the radiusr i and center coordinatesx i, Y i ) are related y the following formula: r i X i Y i +, i 0, ±1, ±,... 1) 4a Note that, for the major circle ordinatey 0, the following relation must hold: ) ) a a Y 0 a a. ) In correspondence of two particular values for Y 0, we have two symmetrical dispositions for the up and down chains: if Y 0 0, we have that r 0 and the major circle is isected y the x-axis central symmetry); if Y 0 ±a a ) ), we have that r 0 a a and two equal major circles, one for the up chain and one for the down chain), oth tangent to x-axis, exist i-central symmetry). 3. Circle radii expressions y inversion In [], the expressions for the center coordinates X i,y i ) and radius r i of the generic i-th circle of the chain have een otained in form of continued fractions. Here we want to get them in a closed form. According to a hint of F. J. García Capítan in a personal communication), such a result can e otained y means of the inversion technique. In particular, we are interested in otaining a closed form formula for the ratio etween the major circle radius and the generick-th circle radius. To this aim, it is worthwhile to remark some points and to write some equations that shall e used in the following: Outer circlec equation: x +y a)x 4a 0 3)
Integer sequences and circle chains inside a circular segment 49 Inversion circle C inv center inh and radiushg) having equation: x +y +4 ay 1a 0 4) and center coordinatesx inv, Y inv ) and radiusr inv given respectively y: X inv 0, Y inv a, R inv 4 a. 5) The inversive image of the y-axis with respect to C inv is still the y-axis while the image of the outer circlec is the straight lines: a )x ay +4a 0. 5) The straight linesis the radical axis relevant to the outer circlec and to the inversion circle C inv see Figure ). For convenience, in Figure we have also added the isector β of the angle etween the y-axis and the straight lines. G β image of the major circle s A O major circle B H C inv Figure. Some circles of the chain with their inversive images The cosine of the angleψ etween they-axis and the straight linesis given y: cosψ a 6) a+ and from 6) one can deduce: sin ψ a+. 7)
50 G. Lucca Now, let us consider, the major circle having center coordinates and radius given respectively y: X 0, Y 0 ) Y 0 ) 4a +, Y 0, 8a) r 0 Y 0 4a +. 8) The inversive image of the major circle, y applying the formulas given in [4], is still a circle having center coordinates and radius given y: X 0, Y 0) 16a Y 0 + Y ) 0 a) 4a +, ) 16a a+ Y 0 +, 9a) a r 0 16a Y 0 + Y ) 0 a) 4a +. 9) The inversive images of the circles inscried in the circular segment are still mutually tangent circles that are also tangent to they-axis and to the straight lines see Figure ); moreover, their centers lie on the isector β. In Figure 3, we show three among these circles: the image of the major circle C 0, the image of the first circle of the up chain C 1 and the image of the first circle of the down chain C 1. In particular y looking at Figure 3, we can easily show, y similitude, that the ratio etween the radius of the generic circlesc i and the radius of the image of the major circlec 0 is: r k r 0 ) 1 sin ψ i sin ψ. 10) From formula 10) and y taking into account 7) and 9), one has: r i X i 1 a+ a+ Y i a 1 a i 16a Y 0 + a) a+ a+ From [4], one can write the following relation: r 0 r i r 0 r i i Y 0 ) 4a +, 11a) Y 0 a)4 Y 0 + + a. 11) a X i X inv) +Y i Y inv) r i ). 1) R inv
Integer sequences and circle chains inside a circular segment 51 y β C 1 s C 0 O C 1 x ψ Figure 3. Inversive images of circles C 1,C 0 andc 1 Such a formula, y taking into account 4), 11a) and 11), and after some algera, ecomes: r 0 r i 1 a+ a+ Y 0 4a) 16a or equivalently, r 0 r i 1 a+ a+ i Y 0 a) + 1 16a i/ a+ a+ Y 0 a) 4 1 a a+ a+ i Y 0 + a) i/ 16a 13a) Y 0 + a) 4 a. 13) Equation 13) simply shows that the ratio r 0 /r i is always a square. Equations 13a) or 13) define the sequence{τ i } {r 0 /r i } fori 0,±1,±,... On this sequence we shall focus our attention in the following sections.
5 G. Lucca 4. A recursive formula for{τ k } If we look at 13a), we note that it is a Binet-like formula of the type: where, in our case, is: τ i Aω i +Bω i +C 14) 1 ω a+ a+, 15) A Y 0 a), B Y 0 + a), C Y 0 4a). 16) 16a 16a 16a In [1], J. Kocik has demonstrated that a generic Binet-like relation can e expressed in an equivalent way y means of a non-homogeneous recursive relation of the type: τ i+ ω + 1 ) τ i+1 τ i +C ω 1 ) 17) ω ω that, y taking into account 15) and the expression forc in 16), ecomes: τ i+ ) [ ) ] 1 Y0 τ i+1 τ i +. 18) a a a 5. Conditions for{τ i } to e an integer sequence In the general case, the sequence {τ i } is composed of real numers. Here we want to find the conditions for which the sequence {τ i } is entirely composed of integers. To this aim, it is convenient to introduce the following variales: u Y 0 a, v with the condition u 1. 19) a The condition for the varialeucomes from formula ). By considering the new variales u and v, equation 13a) ecomes: τ i 1 v v i u v) + 1 v v i u+ v) u 4v). 0) From the general point of view, it is possile to impose, y means of 0), that the ratios τ 1 and τ 1 are equal to two given real numers λ and µ λ > 1 and µ > 1) respectively; that yields the following system: 1 v v v 1 v ) ) u v) + u v) + v 1 v 1 v v ) ) u+ v) u+ v) u 4v) λ, u 4v) µ 1)
Integer sequences and circle chains inside a circular segment 53 having the solution { u λ µ, v λ+ µ) ) 4 1. Thus, if λ m and µ n, m, n eing integers), we have that y choosing a, andy 0 satisfying the relations: Y 0 a m+ n) 4. 3) 4 a m n, we also have that: τ 1 m, τ 1 n. 4) Relations 3) only imply thatτ 1 andτ 1 are integers, ut do not imply thatτ i and τ i,i,3,... are also integers. Nevertheless, if m andnare integers satisfying the following relation: mn K 5) that is their product is equal to a square integer) we have that the coefficients of the recursion 18) are always integer numers and their expressions are: a) +m+k +n, 1 Y0 ) a K +. 6) a Therefore, eing τ 0 1, τ 1 m, τ 1 n and taking into account 6), we have, from 18) that τ and τ too are integers. Finally, due to the fact that the expressions in 6) represent integer numers and eing 18) a recursive relation, we deduce thatτ i is an integer for any indexi, and conclude that 3) together 5) are sufficient conditions in order to e {τ i } an integer sequence. 6. Symmetrical chains Two particular cases relevant to conditions 3) and 5) are represented y: a) m n. )m 1,n K orm K,n 1. In case a) one finds the chains with central symmetry while in case ) the chains with i-central symmetry. In oth cases the up and down sequences are equal and some of them are classified in OEIS The On Line Encyclopedia of Integer Sequences) [3]. We reported them in Tales I and II.
54 G. Lucca Tale I: sequences listed in OEIS and related to chains having central symmetry Values of m and n m n m n 3 m n 4 m n 9 m n 10 Classification of the sequence according to OEIS A055997 A171640 A055793 A05579 A47335 Tale II: sequences listed in OEIS and related to chains having i-central symmetry Classification of the sequence Values of m and n according to OEIS m 1,n 4 or n 1,m 4 A081068 m 1,n 5 or n 1,m 5 A008844 m 1,n 81 or n 1,m 81 A04617 m 1,n 169 or n 1,m 169 A006051 7. Examples Let us show some examples. a) Case of non symmetrical chains: If we choosem and n 8, we have from 3): Y 0 a, a 7. The generated integer sequences up and down are respectively: {τ up } 1,, 5, 39, 641,..., {τ down } 1, 8, 11, 19, 3065,... ) Case of chains with central symmetry: If we choosem and n, we have from 3): Y 0 a 0, a 1. The generated integer sequences up and down are respectively: {τ up } {τ down } 1,, 9, 50, 89, 168,... This is sequence A055997 in OEIS. c) Case of chains with i-central symmetry: If we choosem 1 and n 4, we have from 3): Y 0 a 1, a 5 4.
Integer sequences and circle chains inside a circular segment 55 The generated integer sequences up and down are respectively: {τ up } {τ down } 1, 4, 5, 169, 1156,... This is sequence A081068 in OEIS. References [1] J. Kocik, Lens sequences, arxiv: 0710.36v1[math.NT] 17 Oct. 007; http://arxiv.org/pdf/0710.36.pdf [] G. Lucca, Circle chains inside a circular segment, Forum Geom., 9 009) 173 179. [3] N. J. A. Sloane editor), The On-Line Encyclopedia of Integer Sequences, https://oeis.org. [4] E. W. Weisstein, Inversion, from MathWorld A Wolfram We Resource, http://mathworld.wolfram.com/inversion.html. GIOVANNI LUCCA: VIA CORVI 0, 91 PIACENZA, ITALY E-mail address: V A N N I L U C C A@I N W I N D.I T