Some variations on Fibonacci matrix graphs

Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 23, 2017, No. 4, 85–93 Some variations on Fibonacci matrix graphs Anthony G. Shannon1 and Ömür Deveci2 1 Emeritus Professor, University of Technology Sydney, NSW 2007, Fellow, Warrane College, University of New South Wales, Kensington NSW 2033, Director, Academic Affairs, Australian Institute of Music, Sydney NSW 2010, Australia e-mail: [email protected] 2 Department of Mathematics, Faculty of Science and Letters, Kafkas University 36100, Turkey e-mail: [email protected] Received: 12 March 2016 Accepted: 29 October 2017 Abstract: Matrices are here considered in two ways: arrays containing Fibonacci numbers and their generalizations in the cells, and arrays as graphs where the cells themselves are subgraphs. Both aspects contain ideas for further development and research. Keywords: Fibonacci, Pell and Eulerian numbers, Pyramidal numbers, Golden section, Spanning trees, Lattice points. AMS Classification: 11B39, 05C62. 1 Introduction With a slight variation of Horadam’s classic systematic and simple summary notation [4] we consider aspects of some extensions of some of the well-known sequences in Table 1, though what we elaborate below can be applied to any of these sequences or their generalizations. Horadam and Mahon studied these together with Chebyshev, Gegenbauer, Humbert and Stirling analogues [5]. The plan of this paper is to consider {wk ,n ( a, b; p, q )} for • arbitrary k in Section 2; • fixed k = 2 in Section 3; • variable q in Section 4; • k = 2 & 3 in Section 5. 85 Sequence a b p q wn 0 1 1 –1 Fn Fibonacci 2 1 1 –1 Ln Lucas 0 1 2 –1 Pn Pell 1 3 2 –1 Qn Pell-Lucas 0 1 x+2 +1 Bn(x) Morgan–Voyce Even Fibonacci 1 1 x+2 +1 bn(x) Morgan–Voyce Odd Fibonacci 2 x+2 x+2 +1 Cn(x) Morgan–Voyce Even Lucas -1 1 x+2 +1 cn(x) Morgan–Voyce Odd Lucas 0 1 1 –x Jn(x) Jacobstha–Fibonacci 2 1 1 –x jn(x) Jacobsthal–Lucas 0 1 x +1 Vn(x) Vieta–Fibonacci 2 x x +1 vn(x) Vieta–Lucas Table 1. Integer and polynomial sequences {w2,n ( a, b; p, q )} 2 A Pell variation Consider the kth order Pell generalization {w (1, 2,..., k ; 2, +1)} k ,n formed from the recurrence relation (2.1) uk ,n = 2uk , n −1 − uk ,n − k , n ≥ k , with initial values uk,j = j, j = 0, 1, 2, 3, ... . Some examples are set out in Table 2. k↓, n→ 1 2 3 4 5 6 7 8 9 10 11 12 13 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 3 4 5 6 7 8 9 10 11 12 13 3 1 2 3 5 8 13 21 34 55 89 144 233 377 4 1 2 3 4 7 12 21 38 69 126 231 424 779 5 1 2 3 4 5 9 16 29 54 103 201 393 757 6 1 2 3 4 5 6 11 20 37 70 135 264 517 7 1 2 3 4 5 6 7 13 24 45 86 167 328 8 1 2 3 4 5 6 7 8 15 28 53 102 199 9 1 2 3 4 5 6 7 8 9 17 32 61 118 10 1 2 3 4 5 6 7 8 9 10 19 36 69 Table 2. Patterns among {wk ,n (1,2,..., k ;2,1)} 86 If we consider the elements in the backward diagonals in Table 2, then we can establish the sequences in Table 3 with the connections with the Eulerian numbers: E n = 2 n − n − 1, n ≥ 0. (2.2) k↓, m→ 1 2 3 4 5 6 7 8 9 10 uk,m m> 1 1 1 1 1 1 1 1 1 1 1 u1, m = 1 0 2 1 2 3 4 5 6 7 8 9 10 u 2,m = m 0 3 1 3 5 7 9 11 13 15 17 19 u 3, m = 2 m − 1 1 4 1 4 8 12 16 20 24 28 32 36 u 4,m = 4m − 4 1 5 1 5 13 21 29 37 45 53 61 69 u 5,m = 8m − 11 2 6 1 6 21 38 54 70 86 102 118 134 u 6, m = 16m − 26 3 7 1 7 34 69 103 135 167 199 231 263 u 7 ,m = 32m − 57 4 8 1 8 55 126 201 264 328 392 456 520 u 8,m = 64m − 120 5 9 1 9 89 231 393 517 649 777 905 1033 10 1 10 144 424 757 1014 1290 1546 1802 2058 u 9 ,m = 128m − 247 u10,m = 256m − 502 6 7 Table 3. Eulerian and generalized Pell numbers 3 Fibonacci matrix variations The previous tables suggest realigning the columns by dropping the elements in successive columns to produce Fibonacci rectangular and Lucas square ‘triangular’ matrices as in Tables 4 and 5. F10 × 6 = 1 1 1 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 5 8 1 2 3 0 0 1 0 0 0 0 0 0 1 1 1 1 13 21 34 55 5 8 13 21 2 3 5 8 0 1 2 3 0 0 0 0 Table 4. A Fibonacci triangular matrix 87 Thus, F5×1 = [1, 1, 1, 1, 1]T for instance. The ‘left triangle’ parts of these matrices lack the symmetry that one finds with Pascal-type triangles of these sorts of numbers [11]. Nevertheless, the Fibonacci triangle set out in Table 4 has several properties similar to these generalizations [9], including the following examples with a variety of row, column and diagonal properties can be discerned. Sums of cells: • row sums are Fibonacci numbers; • partial row sums {Fn +3 − 2}, j > 1; • rising diagonal sums {Fn +1 − 1 2 Fn+1 }. Partial recurrence relations: • u i , j = u i −1, j + u i − 2 , j , j > 1; • u i , j = u i , j −1 − u i −1, j −1 , i > 1, j > i − 1; • u i , j = Fi − 2 j + 4 . Analogous variations can also be applied to the other sequences to produce companion matrices [3] and tridiagonal matrices [2]. Instead we now outline a corresponding Lucas illustration. L10×10 = 1 1 1 0 2 1 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 4 7 1 3 4 2 1 3 0 2 1 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 11 18 29 47 7 11 18 29 4 7 11 18 3 4 7 11 1 3 4 7 2 1 3 4 0 2 1 3 0 0 2 1 0 0 0 1 Table 5. A Lucas triangular matrix 4 Golden ratio variations Variations of the golden ratio are effectively done by looking at the sequences generated with different values of q. A Fibonacci golden ratio family of sequences is set out in Table 6 [cf. 10], and its Lucas counterpart in Table 7 [cf. 17]. Thus one can use Sloane’s encyclopedia for connections and creations [15]. More deeply one can search for intersections [16] and divisibility properties [7]. 88 wn (0,1;1,−q) 1 2 3 4 5 6 7 8 9 10 11 12 wn (0,1;1,0) 1 1 1 1 1 1 1 1 1 1 1 1 wn (0,1;1,−1) 1 1 2 3 5 8 13 21 34 55 89 144 wn (0,1;1,−2) 1 1 3 5 11 21 43 85 171 341 683 1365 wn (0,1;1,−3) 1 1 4 7 19 40 97 217 508 1159 2683 6160 wn (0,1;1,−4) 1 1 5 9 29 65 181 441 1165 2929 7589 19305 wn (0,1;1,−5) 1 1 6 11 41 96 301 781 2286 6191 17621 48576 wn (0,1;1,−6) 1 1 7 13 55 133 463 1261 4039 11605 35839 105469 wn (0,1;1,−7) 1 1 8 15 71 176 673 1905 6616 19951 66263 205920 Table 6. Fibonacci variations wn (2,1;1,− q ) 0 1 2 3 4 5 6 7 8 9 10 11 wn (2,1;1,0) 2 2 2 2 2 2 2 2 2 2 2 2 wn (2,1;1,−1) 2 1 3 4 7 11 18 29 47 76 123 199 wn (2,1;1,−2) 2 1 5 7 17 31 65 127 257 511 1025 2047 wn (2,1;1,−3) 2 1 7 10 31 61 154 337 799 1810 4207 9637 wn (2,1;1,−4) 2 1 9 13 49 101 297 701 1889 4693 12249 31021 wn (2,1;1,−5) 2 1 11 16 71 151 506 1261 3791 10096 29051 79531 wn (2,1;1,−6) 2 1 13 19 97 211 793 2059 6817 19171 60073 175099 wn (2,1;1,−7) 2 1 15 22 127 281 1170 3137 11327 33286 112575 345577 Table 7. Lucas variations 5 Cells in matrices We now consider the matrix arrays as graphs in themselves. For simplicity, we start with square matrices which are divided into sub-graphs containing 1, 4, 9, 16, ..., square matrices as illustrated in Figure 1. Figure 1. Matrices with 1, 4, 9, 16 cells Immediately we observe that the number of squares contained in each matrix is 1, 5, 14, 30, 55, ..., the square pyramidal numbers, generated by n(n + 1)(2n + 1)/6, where n2 is the number of cells contained with the whole matrix. There is a wealth of literature on pyramidal numbers [17: M3844] which we do not plan to pursue here. Rather we continue to consider aspects of these subgraphs. 89 Connections with Fibonacci matrices and graphs occur through spanning trees and the complexity of a graph [6, 12, 14], but many problems remain. For instance, by extending the squares in Figure 1 through their diagonals, we obtain the planar representation of a trellis (or wire mesh) fence consists of sets of ‘crosses’ or ‘squares’ as shown in Figure 2. Figure 2. Representation of a section of trellis (wire-mesh) Immediately a number of non-trivial questions arise, such as how many squares? symmetric crosses? rectangles? lattice points? crosses (symmetric or asymmetric)? spanning trees? Attempts to solve the problems are probably best illustrated by construction. In general, one would expect the solutions to be functions of the numbers of edges and vertices. We define a trellis of a given size and position in the plane as even or odd: • an even trellis, fn,m, is the set of integer lattice points {( x, y ) : x + y is even, 0 ≤ x ≤ 2n, 0 ≤ y ≤ 2m}; an odd trellis, gn,m, is the set of integer lattice points {( x, y ) : x + y is odd, 0 ≤ x ≤ 2n, 0 ≤ y ≤ 2m}. Figures 3 (a), (b), (c), (d) show the cases for ‘fences’ f1,m, f2,m, f3,m , m = 1, 2, 3, respectively, where {fn,m} represents the set of single-edged symmetric crosses of fences with ‘height’ n and ‘length’ m. Thus in Figure 2, {f2,m}, m = 1, 2, 3, is the set of 3 single-edged symmetric crosses of height 2 such crosses. • (a) f 1,m (b) f 2,m (c) f 3,m (d) g 3, 2 Figure 3. Representation of fences 90 Let en,m be the number of edges in fn,m. Then, since fn,m is constructed by an n × m lattice of crosses and each cross contributes four edges, it follows that fn,m = 4nm. (5.1) See the black dots in Figure 4 and the entries in Table 8. Figure 4. f 2 ,3 n↓ m→ 1 2 3 4 1 4 8 12 16 2 8 16 24 32 3 12 24 36 48 4 16 32 48 64 5 20 40 60 80 Table 8. e n ,m Similarly let v n ,m ∈ f n ,m and wn ,m ∈ g n ,m be the corresponding numbers of vertices (Table 9). n↓ m→ 1 2 3 4 1 5 8 11 14 2 8 13 18 23 3 11 18 25 32 4 14 23 32 41 5 17 28 39 50 Table 9. v n ,m For v n ,m there are nm black dots and (n+1)(m+1) white dots for a total of v n , m = nm + (n + 1)(m + 1) = 2nm + n + m + 1. 91 (5.2) Figure 5. g3,6 6 Concluding comments Various other extensions and generalizations of the sequences in Table 1 can be readily investigated. For example, just as for the second-order Pell sequences {P } ≡ {w (0,1;2,−1)}, {Q } ≡ {w (1,3;2,−1)} 2,n 2,n 2,n 2 ,n there is the connection (4.1) Q2 ,n = P2,n + P2,n +1 so too for the corresponding third-order Pell sequences {P } ≡ {w (0,0,1;2,−1)}, {Q } ≡ {w (1,1,3;2,−1)} 3,n 3, n 3, n 3, n there is also the connection Q3, n = P3,n −1 + P3, n + P3,n +1 (5.2) where the third-order recurrence relation is w3,n = w3,n −1 + w3,n − 2 , n ≥ 2. While at one level almost any desired elegant identity can be obtained by a suitable choice of initial values, the selection can be determined by us with the use of “basic” sequences and corresponding matrices. At order k, there will be k basic fundamental sequences and one primordial sequence, and corresponding matrices [13]. 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