Quasi-Fibonacci Numbers of the Seventh Order

1 2 3 4 6 23 11 Journal of Integer Sequences, Vol. 9 2006, Article 06.4.3 Quasi-Fibonacci Numbers of the Seventh Order Roman Witu la, Damian S lota and Adam Warzyńsi Institute of Mathematics Silesian University of Technology Kaszubsa 23 Gliwice 44-100 Poland r.witula@polsl.pl d.slota@polsl.pl Abstract In this paper we introduce and investigate the so-called quasi-fibonacci numbers of the seventh order. We discover many surprising relations and identities, and study some applications to polynomials. 1 Introduction Grzymowsi and Witu la [3] discovered and studied the following two identities: 1 ξ ξ 4 n F n1 F n ξ ξ 4, 1.1 1 ξ 2 ξ 3 n F n1 F n ξ 2 ξ 3, 1.2 where F n denote the Fibonacci numbers and ξ C is a primitive fifth root of unity i.e., ξ 5 1 and ξ 1. These identities mae it possible to prove many classical relations for Fibonacci numbers as well as to generalize some of them. We may state that these identities mae up an independent method of proving such relations, which is an alternative to the methods depending on the application of either Binet formulas or the generating function of Fibonacci and Lucas numbers. 1

1.1 Example of the application of identities 1.1 and 1.2 First, we note that 1 ξ ξ 4 uv F uv1 F uv ξ ξ 4 1.3 and 1 ξ ξ 4 uv 1 ξ ξ 4 u 1 ξ ξ 4 v F u1 F u ξ ξ 4 F v1 F v ξ ξ 4 by the identity 1 ξ ξ 2 ξ 3 ξ 4 0 F u1 F v1 F u F v F u F v1 F u1 F v F u F v ξ ξ 4 Replacing u by u r and v by v r in 1.4 we obtain 1 ξ ξ 4 uv F u r1 F vr1 F u r F vr F u1 F v1 F u F v F u F v1 F u 1 F v ξ ξ 4. 1.4 F u r F vr1 F u r1 F vr F u r F vr ξ ξ 4. 1.5 We note that the numbers 1 and ξ ξ 4 are linearly independent over Q. Hence comparing the parts without ξξ 4 of 1.3 with 1.4 and 1.4 with 1.5 we get two nown identities see [2, 5] F uv1 F u1 F v1 F u F v and F u1 F v1 F u r1 F vr1 F u r F vr F u F v after the next u r iterations 1 u r1 F r F v ur F 0 F v u2r 1 u r1 F r F v ur. 1.2 The aim of the paper In this paper the relations 1.1 and 1.2 will be generalized in the following way: 1 δξ ξ 6 n A n δ B n δξ ξ 6 C n δξ 2 ξ 5, 1 δξ 2 ξ 5 n A n δ B n δξ 2 ξ 5 C n δξ 3 ξ 4, and 1 δξ 3 ξ 4 n A n δ B n δξ 3 ξ 4 C n δξ ξ 6, where ξ C are primitive seventh roots of unity i.e., ξ 1 and ξ 1, δ C, δ 0. New families of numbers created by these identities {A n δ} n1, {B n δ} n1, and {C n δ} n1 1.6 2

called here the quasi-fibonacci numbers of order ;δ, δ C, δ 0, are investigated in this paper. The elements of each of the three sequences 1.6 satisfy the same recurrence relation of order three X n3 δ 3X n2 3 2δ 2δ 2 X n1 1 δ 2δ 2 δ 3 X n 0, which enables a direct trigonometrical representation of these numbers see formulas 3.1 3.19. In consequence, many surprising algebraic and trigonometric identities and summation formulas for these numbers may be generated. Also the polynomials connected with the numbers {C n δ} n1 C 1 δx n, n N, are investigated in this paper. 1 2 Minimal polynomials, linear independence over Q Let Ψ n x be the minimal polynomial of cos2π/n for every n N. W. Watins and J. Zeitlin described [10] see also [9] the following identities: T s1 x T s x 2 s d n Ψ d x if n 2s 1 and T s1 x T s 1 x 2 s d n Ψ d x if n 2s, where T s x denotes the s-th Chebyshev polynomial of the first ind. In the sequel, if n 2s 1 is a prime number, we obtain For example, we have T s1 x T s x 2 s Ψ 1 xψ n x. 1 Ψ x 8x 1 T 1 4x T 3 x 8x 1 8x4 4x 3 8x 2 3x 1 1 8x 1 x 3 1 8x 1 2 x2 1 2 x 1 x 3 1 8 2 x2 1 2 x 1 8, Lemma 2.1. If n 3 then the roots of Ψ n x 0 are cos2π/n, for 0 < s and,n 1 where n 2s or n 2s 1 respectively. Below a more elementary proof of this result but only for the polynomial Ψ x, will be presented. Our proof is based only on the following simple fact: } Lemma 2.2. Let q Q and cosqπ Q. Then cosqπ {0, ± 12, ±1. 3

Proof. For the elementary proof of this Lemma see, for example, the Appendix to the Russian translation of Niven s boo [6] written by I. M. Yaglom. Lemma 2.3. Let ξ expi2π/. Then the polynomial p x : x ξ ξ 6 x ξ 2 ξ 5 x ξ 3 ξ 4 x 3 x 2 ξ ξ 2 ξ 3 ξ 4 ξ 5 ξ 6 xξ ξ 6 ξ 2 ξ 5 ξ ξ 6 ξ 3 ξ 4 ξ 2 ξ 5 ξ 3 ξ 4 ξ ξ 6 ξ 2 ξ 5 ξ 3 ξ 4 x 3 x 2 2x 1 is a minimal polynomial of the numbers ξ ξ 6 2 cos2π/, ξ 2 ξ 5 2 cos4π/, ξ 3 ξ 4 2 cos6π/. 2.1 Moreover, we have the identity Ψ x 1 8 p 2x. 2.2 Proof. Besides Lemma 2.2, each of the numbers in 2.1 is an irrational number, so the polynomial p x is irreducible over Q. Corollary 2.4. We have p x 1 x 3 2x 2 x 1. See the identity 3.66 below, where the connection of p x 1 with quasi-fibonacci numbers of -th order is presented. Corollary 2.5. The numbers cos2π/, 0, 1, 2 are linearly independent over Q. Proof. If we suppose that for some a,b,c Q, then we also have a b cos2π/ c cos4π/ 0 a b cos2π/ c2 cos 2 2π/ 1 0, i.e., a c b cos2π/ 2c cos 2 2π/ 0 so the degree of cos2π/ is 2, which by Lemma 2.3 means that a b c 0. Corollary 2.6. Every three numbers which belong to the set {1,ξ ξ 6,ξ 2 ξ 5,ξ 3 ξ 4 } are linearly independent over Q. Proof. It follows from the identity 1 ξ ξ 2... ξ 6 0 and from Corollary 2.5. 4

Corollary 2.. The following decomposition holds: f n X : X 2 cos 2π 2 n X 2 cos 4π 2 n X 3 α n X 2 β n X 1, where α 0 1, β 0 2 and X 2 cos 6π 2 n { αn1 α 2 n 2β n, β n1 β 2 n 2α n, n N. 2.3 For example, we have α 0 1, β 0 2, α 1 5, β 1 6, α 2 13, β 2 26, α 3 11 and β 3 650. A more general decomposition will be presented later, see Lemma 3.14 a. We note that 13 α n and 13 β n for every n N, n 2. Proof. We have Y 2 2 cos 2π 2 n1 Y 2 2 cos 4π 2 n1 Y 2 2 cos 6π 2 n1 f n Y f n Y Y 3 α n Y 2 β n Y 1Y 3 α n Y 2 β n Y 1 Y 3 β n Y 2 α n Y 2 1 2 Y 6 α 2 n 2β n Y 4 β 2 n 2α n Y 2 1. It is obvious that α n 2 cos 2π 2 n 2 cos 4π 2 n 2 cos 6π 2 n and β n 4 cos 2π cos 4π 2 n 4 cos 2π cos 6π 2 n 4 cos 4π cos 6π 2 n for every n N. Moreover, from 2.3, we deduce that which yields α n1 β n1 α n 1 2 βn 1 2 2, α n1 β n1 α n 1 2 βn 1 2 αn β n αn β n 2 α n β n αn 1 1 2 βn 1 1 2, n 1 α β n1 α n1 1 2 β 1 n 1 2 α 13 1 2 β 1 2 n α β 2. 1 The following extension lemma is the basic technical tool to generating almost all formulas presented in the next sections. 5 1

Lemma 2.8. Let a 1,a 2,...,a n R be linearly independent over Q and let f,g Q[δ], 1, 2,...,n. If the identity f δa 1 g δa 1 holds for every δ Q, then f δ g δ for every 1, 2,...,n and δ C. 3 Quasi-Fibonacci numbers of order Lemma 3.1. Let δ,ξ C, ξ 1 and ξ 1. Then the following identities hold: 1 δξ ξ 6 n a n δ b n δξ 1 1 ξ 1 6 c n δξ 2 1 ξ 2 6 d n δξ 3 1 ξ 3 6 a n δ b n δξ ξ 6 c n δξ 2 ξ 5 d n δξ 3 ξ 4 A n δ B n δξ ξ 6 C n δξ 2 ξ 5, 3.1 1 δξ 2 ξ 5 n a n δ b n δξ 1 2 ξ 1 5 c n δξ 2 2 ξ 2 5 d n δξ 3 2 ξ 3 5 a n δ b n δξ 2 ξ 5 c n δξ 3 ξ 4 d n δξ ξ 6 A n δ B n δξ 2 ξ 5 C n δξ 3 ξ 4, 3.2 1 δξ 3 ξ 4 n a n δ b n δξ 1 3 ξ 1 4 c n δξ 2 3 ξ 2 4 d n δξ 3 3 ξ 3 4 a n δ b n δξ 3 ξ 4 c n δξ ξ 6 d n δξ 2 ξ 5 A n δ B n δξ 3 ξ 4 C n δξ ξ 6, 3.3 where a 0 δ 1, b 0 δ c 0 δ d 0 δ 0, a n1 δ a n δ 2δb n δ, b n1 δ δa n δ b n δ δc n δ, c n1 δ δb n δ c n δ δd n δ, d n1 δ δc n δ 1 δd n δ 3.4 6

for every n N and A n1 δ : a n1 δ d n1 δ a n δ 2δb n δ δc n δ 1 δd n δ a n δ d n δ 2δb n δ d n δ δc n δ d n δ A n δ 2δB n δ δc n δ, B n1 δ : b n1 δ d n1 δ δa n δ b n δ δc n δ δc n δ 1 δd n δ δa n δ d n δ b n δ d n δ δa n δ B n δ, C n1 δ : c n1 δ d n1 δ δb n δ c n δ δd n δ δc n δ 1 δd n δ δb n δ d n δ 1 δc n δ d n δ δb n δ 1 δc n δ, i.e., for every n N. A 0 δ 1, B 0 δ C 0 δ 0, A n1 δ A n δ 2δB n δ δc n δ, B n1 δ δa n δ B n δ, C n1 δ δb n δ 1 δc n δ 3.5 Proof. For example, we have 1 δξ ξ 6 n1 1 δξ ξ 6 n 1 δξ ξ 6 a n δ b n δξ ξ 6 c n δξ 2 ξ 5 d n δξ 3 ξ 4 1 δξ ξ 6 a n δ δa n δξ ξ 6 b n δξ ξ 6 δb n δξ 2 2 ξ 5 c n δξ 2 ξ 5 δc n δξ 3 ξ ξ 6 ξ 4 d n δξ 3 ξ 4 δd n δξ 4 ξ 5 ξ 2 ξ 3 a n δ 2δb n δ ξ ξ 6 δa n δ b n δ δc n δ ξ 2 ξ 5 δb n δ c n δ δd n δ ξ 3 ξ 4 δc n δ 1 δd n δ by equality ξ 3 ξ 4 1 ξ ξ 6 ξ 2 ξ 5 a n δ 2δb n δ δc n δ 1 δd n δ ξ ξ 6 δa n δ b n δ 1 δd n δ ξ 2 ξ 5 δb n δ 1 δc n δ d n δ. Definition 3.2. The numbers 3.5 will be called the quasi-fibonacci numbers of order ; δ see Table 1 at the end of the paper. To simplify notation we write a n, b n, c n, d n, A n, B n, C n instead of a n 1, b n 1, c n 1, d n 1, A n 1, B n 1, C n 1 respectively, and these numbers will be called the quasi-fibonacci numbers of the seventh order.

Remar 3.3. Note that section III of [8] contains an explicit reference to the sequences A n, B n and C n. This paper is partly based on the properties of p x and p x 1. A n, B n and C n also appear in [4]. Corollary 3.4. Adding equalities 3.1 3.3 we obtain the identity 1 δξ ξ 6 n 1 δξ 2 ξ 5 n 1 δξ 3 ξ 4 n 3A n δ B n δ C n δ. 3.6 Corollary 3.5. There follows from 3.4 and 3.5 two special systems for δ 1 a 0 1, b 0 c 0 d 0 0, a n1 a n 2b n, b n1 a n b n c n, c n1 b n c n d n, d n1 c n 2d n 3. and for every n N see Table 2, A006356 and A006054 in []. A 0 1, B 0 C 0 0, A n1 A n 2B n C n, B n1 A n B n, C n1 B n 3.8 Corollary 3.6. Another system may also be derived from 3.5 for δ 1 A 0 1 1, B 0 1 C 0 1 0, A n1 1 A n 1 2B n 1 C n 1, B n1 1 A n 1 B n 1, C n1 1 B n 1 2C n 1 for every n N see Table 3 and A085810 in []. Corollary 3.. If δ 0 then from 3.5 the following identities can be generated: hence, we obtain δb n δ C n1 δ 1 δc n δ, 3.9 δa n δ B n1 δ B n δ, 3.10 δ 2 A n δ δb n1 δ δb n δ 3.11 C n2 δ 1 δc n1 δ C n1 δ 1 δc n δ C n2 δ 2 δc n1 δ 1 δc n δ, A n1 δ A n δ 2δB n δ δc n δ 3.12 2C n1 δ 21 δc n δ δc n δ 2C n1 δ 2 δc n δ, δ 2 A n1 δ δ 2 A n δ 2δ 2 C n1 δ 2δ 2 δ 3 C n δ. 3.13 8

On the other hand, by 3.11 we obtain δ 2 A n1 δ δ 2 A n δ C n3 δ 2 δc n2 δ 1 δc n1 δ C n2 δ 2 δc n1 δ 1 δc n δ C n3 δ 3 δc n2 δ 3 2δC n1 δ δ 1C n δ. 3.14 From 3.13 and 3.14 we obtain the final recurrence identities for numbers C n δ C n3 δ δ 3C n2 δ 3 2δ 2δ 2 C n1 δ 1 δ 2δ 2 δ 3 C n δ 0. 3.15 Remar 3.8. We note that the elements of each of the sequences {A n δ} and {B n δ} also satisfy this recurrence relation which follows from 3.9 and 3.10. Lemma 3.9. The characteristic polynomial p X;δ of 3.15 has the following decomposition: p X;δ : X 3 δ 3X 2 3 2δ 2δ 2 X 1 δ 2δ 2 δ 3 where ξ expi2π/. So the relation X 1 δξ ξ 6 X 1 δξ 2 ξ 5 X 1 δξ 3 ξ 4 3.16 C n δ α1 δξ ξ 6 n β1 δξ 2 ξ 5 n γ1 δξ 3 ξ 4 n holds for every n N and for some α,β,γ R. Remar 3.10. To find α,β,γ, from Lemma 3.9, it is sufficient to solve the following linear system: C 1 δ 0 α1 δξ ξ 6 β1 δξ 2 ξ 5 γ1 δξ 3 ξ 4, C 2 δ δ 2 α1 δξ ξ 6 2 β1 δξ 2 ξ 5 2 γ1 δξ 3 ξ 4 2, C 3 δ 3δ 2 δ 3 α1 δξ ξ 6 3 β1 δξ 2 ξ 5 3 γ1 δξ 3 ξ 4 3. After some calculations we obtain the identities C n δ ξ 2 1 ξ 2 6 ξ 3 1 ξ 3 6 1 δξ ξ 6 n 3.1 ξ 2 2 ξ 2 5 ξ 3 2 ξ 3 5 1 δξ 2 ξ 5 n ξ 2 3 ξ 2 4 ξ 3 3 ξ 3 4 1 δξ 3 ξ 4 n ξ 2 ξ 5 ξ 3 ξ 4 1 δξ ξ 6 n ξ 3 ξ 4 ξ ξ 6 1 δξ 2 ξ 5 n ξ ξ 6 ξ 2 ξ 5 1 δξ 3 ξ 4 n 2 cos 4 π cos π 1 2δ cos 2π n 2 cos π cos 2 π 1 2δ cos 4π n 2 cos 2π cos 4π 1 2δ cos 6π n. 9

Hence, by 3.9 we get and by 3.10 B n δ 1 δ C n1δ 1 δc n δ 3.18 ξ 1 1 ξ 1 6 ξ 3 1 ξ 3 6 1 δξ ξ 6 n ξ 1 2 ξ 1 5 ξ 3 2 ξ 3 5 1 δξ 2 ξ 5 n ξ 1 3 ξ 1 4 ξ 3 3 ξ 3 4 1 δξ 3 ξ 4 n ξ ξ 6 ξ 3 ξ 4 1 δξ ξ 6 n ξ 2 ξ 5 ξ ξ 6 1 δξ 2 ξ 5 n ξ 3 ξ 4 ξ 2 ξ 5 1 δξ 3 ξ 4 n 2 cos 2π cos π 1 2δ cos 2π n 2 cos 4π cos 2π 1 2δ cos 4π n 2 cos π cos 4π 1 2δ cos 6π n A n δ 1 δ B n1δ B n δ ξ 0 1 ξ 0 6 ξ 3 1 ξ 3 6 1 δξ ξ 6 n ξ 0 2 ξ 0 5 ξ 3 2 ξ 3 5 1 δξ 2 ξ 5 n ξ 0 3 ξ 0 4 ξ 3 3 ξ 3 4 1 δξ 3 ξ 4 n 2 ξ 3 ξ 4 1 δξ ξ 6 n 2 ξ ξ 6 1 δξ 2 ξ 5 n 2 ξ 2 ξ 5 1 δξ 3 ξ 4 n 2 1 cos π 1 2δ cos 2π n 2 1 cos 2π 1 2δ cos 4π n 2 1 cos 4π 1 2δ cos 6π n. 3.19 Remar 3.11. In the sequel, for δ 1, we obtain from 3.15 C n3 2C n2 C n1 C n 0 3.20 for every n N. Of course, the elements of the sequences {B n } and {A n } satisfy the identity 3.20. The characteristic polynomial of 3.20 has the following decomposition: see Corollary 2.4 X 3 2X 2 X 1 X 1 ξ ξ 6 X 1 ξ 2 ξ 5 X 1 ξ 3 ξ 4, where ξ expi2π/. Moreover, the decompositions of C n there are cyclic transformations of the sums ξ ξ 6, ξ 2 ξ 5 and ξ 3 ξ 4 in elements of the first equality follows immediately from 3.1 C n ξ 2 ξ 5 ξ 3 ξ 4 1 ξ ξ 6 n ξ 3 ξ 4 ξ ξ 6 1 ξ 2 ξ 5 n ξ ξ 6 ξ 2 ξ 5 1 ξ 3 ξ 4 n ξ ξ 6 ξ 3 ξ 4 1 ξ ξ 6 n 1 ξ 2 ξ 5 ξ ξ 6 1 ξ 2 ξ 5 n 1 for every n 1, 2, 3,... 10 ξ 3 ξ 4 ξ 2 ξ 5 1 ξ 3 ξ 4 n 1 3.21

Lemma 3.12. The following summation formulas for the elements of the sequences {C n δ}, {B n δ} and {A n δ} hold: hence, from 3.12 we obtain δ δ N A n δ B N1 δ δ; 3.22 n1 N B n δ C N1 δ δ n1 N C n δ; 3.23 n1 i.e., N N N A N1 δ A 1 δ 2δ B n δ δ C n δ 2C N1 δ δ C n δ, 3.24 n1 n1 n1 N δ C n δ 1 A N1 δ 2C N1 δ; 3.25 n1 δ N C n δ C N1 δ B N1 δ 3.26 n1 3 2 A N1 δ 2 δ δ A N2δ; N δ 3 C n δ C N3 δ δ 2C N2 δ 1 δ 2δ 2 C N1 δ δ 2 and n1 C N2 δ δ 2C N1 δ δ 3 2δ 2 δ 1C N δ δ 2 ; 3.2 δ N B n δ 1 A N1 δ C N1 δ 3.28 n1 N δ 3 B n δ C N3 δ δ 2C N2 δ 1 δ δ 2 C N1 δ δ 2. 3.29 n1 Definition 3.13. We define A n δ : 3A n δ B n δ C n δ, δ C, n N. Moreover, we set A n : A n 1 see Table and A033304 in []. Lemma 3.14. From identities 3.1 3.19 the following special identities can be deduced: a A 2 n A 2n 2ξ ξ 6 n 2ξ 2 ξ 5 n 2ξ 3 ξ 4 n A 2n 2 2 cos 2π n 2 2 cos 4π n 2 2 cos 6π n 11

or 2 cos 2π and 4 cos 2π cos 4π n 2 cos 4π n 2 cos 6π n 1 2 A2 n A 2n, n 4 cos 2π cos 6π n 4 cos 4π cos 6π n 1 n[ 1 ξ ξ 6 n 1 ξ 2 ξ 5 n 1 ξ 3 ξ 4 n] 1 n A n. Consequently, we get the following decomposition: X 2 cos 2π n X 2 cos 4π n X 2 cos 6π n X 3 1 2 A2 n A 2n X 2 1 n A n X 1; b 1 2 n A n 2 cos π 2n cos 2π 2n cos 3π 2n; c A 3 n A 3n 6 1 n 3A n 1 3ξ ξ 6 ξ 2 ξ 5 1 n 3ξ 3 ξ 4 ξ ξ 6 1 n 3ξ 2 ξ 5 ξ 3 ξ 4 1 n ; d e C n1 A n C n A n1 1 ξ 3 ξ 4 4ξ 2 ξ 5 ξ 3 ξ 4 n 1 ξ 2 ξ 5 4ξ ξ 6 ξ 2 ξ 5 n 1 ξ ξ 6 4ξ 3 ξ 4 ξ ξ 6 n 1 2 cos 6π 8 cos 4π 2 cos 6π n 1 2 cos 4π 8 cos 2π 2 cos 4π n ; for n 2, 4; 1 2 cos 2π 8 cos 6π 2 cos 2π n ; for n 6; 0; for n 8; 14; for n 1; A n1/2 ; for n 3, 5, ; B n1 C n1 2A n1 A n B n C n B n C n 2A n A n1 B n1 C n1 8 3ξ 2 ξ 5 ξ ξ 6 n 8 3ξ ξ 6 ξ 3 ξ 4 n 8 3ξ 3 ξ 4 ξ 2 ξ 5 n 8 6 cos 4π 2 cos 2π n 8 6 cos 2π 2 cos 6π n 8 6 cos 6π 2 cos 4π n 14; for n 1; 49; for n 2; 56; for n 3, 12

and a more general identity f αa n1 βb n1 γc n1 εa n ωb n ϕc n αa n βb n γc n εa n1 ωb n1 ϕc n1 α 0 β 0 ξ ξ 6 γ 0 ξ 2 ξ 5 1 ξ ξ 6 n1 α 0 β 0 ξ 2 ξ 5 γ 0 ξ 3 ξ 4 1 ξ 2 ξ 5 n1 α 0 β 0 ξ 3 ξ 4 γ 0 ξ ξ 6 1 ξ 3 ξ 4 n1 ε 0 ω 0 ξ ξ 6 ϕ 0 ξ 2 ξ 5 1 ξ ξ 6 n where ε 0 ω 0 ξ 2 ξ 5 ϕ 0 ξ 3 ξ 4 1 ξ 2 ξ 5 n ε 0 ω 0 ξ 3 ξ 4 ϕ 0 ξ ξ 6 1 ξ 3 ξ 4 n α 0 β 0 ξ ξ 6 γ 0 ξ 2 ξ 5 1 ξ ξ 6 n α 0 β 0 ξ 2 ξ 5 γ 0 ξ 3 ξ 4 1 ξ 2 ξ 5 n α 0 β 0 ξ 3 ξ 4 γ 0 ξ ξ 6 1 ξ 3 ξ 4 n ε 0 ω 0 ξ ξ 6 ϕ 0 ξ 2 ξ 5 1 ξ ξ 6 n1 ε 0 ω 0 ξ 2 ξ 5 ϕ 0 ξ 3 ξ 4 1 ξ 2 ξ 5 n1 ε 0 ω 0 ξ 3 ξ 4 ϕ 0 ξ ξ 6 1 ξ 3 ξ 4 n1 A ξ ξ 6 B ξ 2 ξ 5 C ξ ξ 6 n A ξ ξ 6 C ξ 3 ξ 4 B ξ 3 ξ 4 n A ξ 2 ξ 5 B ξ 3 ξ 4 C ξ 3 ξ 5 n A 2 cos 2π B 2 cos 4π C 2 cos 2π n A 2 cos π B 2 cos 2π C 2 cos π n A 2 cos 4π B 2 cos π C 2 cos 4π n, A : 5ε 0 β 0 5α 0 ω 0 α 0 ϕ 0 ε 0 γ 0 2ω 0 γ 0 2β 0 ϕ 0, B : 5α 0 ω 0 ε 0 β 0 3ε 0 γ 0 3α 0 ϕ 0 γ 0 ω 0 β 0 ϕ 0, C : 2ε 0 β 0 2α 0 ω 0 2β 0 ϕ 0 2γ 0 ω 0 ε 0 γ 0 α 0 ϕ 0, α 0 : 3α β γ, β 0 : α 2β γ, γ 0 : α β 2γ, ε 0 : 3ε ω ϕ, ω 0 : ε 2ω ϕ, ϕ 0 : ε ω 2ϕ. Proof. The identity b follows from 3.6. 13

Remar 3.15. The sequence { 1 2 A2 n A 2n } n1 see Table is an accelerator sequence for Catalan s constant see [1]. The generating function of these numbers has the form see also A094648 in []. 3 2x 2x 2 1 x 2x 2 x 3 Some applications of identities 3.1 3.3 will be presented now. 3.1 Reduction formulas for indices Lemma 3.16. The following identities hold: Proof. First, we note that A mn δ A m δa n δ 2B m δb n δ C m δc n δ 3.30 B m δc n δ B n δc m δ, B mn δ C m δc n δ A m δb n δ A n δb m δ, 3.31 C mn δ B m δb n δ C m δc n δ A m δc n δ 3.32 A n δc m δ B m δc n δ B n δc m δ. 1 δξ ξ 6 mn A mn δ B mn δξ ξ 6 C mn δξ 2 ξ 5. On the other hand, we have 1δξ ξ 6 mn 1 δξ ξ 6 m 1 δξ ξ 6 n A m δ B m δξ ξ 6 C m δξ 2 ξ 5 A n δ B n δξ ξ 6 C n δξ 2 ξ 5 A m δa n δ B m δb n δξ 2 ξ 5 2 C m δc n δξ 4 ξ 3 2 A m δb n δξ ξ 6 A m δc n δξ 2 ξ 5 A n δb m δξ ξ 6 A n δc m δξ 2 ξ 5 B m δc n δξ ξ 6 ξ 2 ξ 5 B n δc m δξ ξ 6 ξ 2 ξ 5 we have ξ ξ 6 ξ 2 ξ 5 ξ 3 ξ 6 ξ ξ 4 1 ξ 2 ξ 5 A m δa n δ 2B m δb n δ C m δc n δ B m δc n δ B n δc m δ ξ ξ 6 A m δb n δ A n δb m δ C m δc n δ ξ 2 ξ 5 B m δb n δ C m δc n δ A m δc n δ A n δc m δ B n δc m δ B m δc n δ hence by linear independence of 1, ξ ξ 6 2 cos 2π, ξ2 ξ 5 2 cos 4π Lemma 2.8 the reduction formulas follow. over Q and by 14

Corollary 3.1. We have A 2n δ A 2 nδ B 2 nδ B n δ C n δ 2, 3.33 B 2n δ 2A n δb n δ C 2 nδ, 3.34 C 2n δ B 2 nδ C 2 nδ 2C n δa n δ B n δ. 3.35 Remar 3.18. Which comes from [], sequence A006356 Let u, v, w be defined by u1 1, v1 0, w1 0 and Then we have u 1 u v w, v 1 u v, w 1 u. un 1 A n, vn 1 B n, wn 1 B n C n ; for every n N {0}. So, according to Table 2 we get {un} 1, 1, 3, 6, 14, 31,..., {vn} 0, 1, 2, 5, 11, 25,..., and {wn} {un} Benoit Cloitre abcloitre@wanadoo.fr, Apr 05 2002. Moreover u 2 v 2 w 2 u2 Gary Adamson, Dec 23 2003. The n-th term of the sequence {un} is the number of paths for a ray of light that enters two layers of glass, and then is reflected exactly n times before leaving the layers of glass. One such path with 2 plates of glass and 3 reflections might be...\.../... -----------------...\/\.../... -----------------...\/... ----------------- 15

Corollary 3.19. We have A 3n δ A n δa 2n δ B n δc 2n δ B 2n δc n δ 3.36 2B n δb 2n δ C n δc 2n δ A 3 nδ B 3 nδ 6A n δb 2 nδ 3A n δc 2 nδ 3B n δc 2 nδ 3B 2 nδc n δ 6A n δb n δc n δ, B 3n δ A 2n δb n δ A n δb 2n δ C n δc 2n δ 3.3 2B 3 nδ C 3 nδ 3A 2 nδb n δ 3A n δc 2 nδ 3B n δc 2 nδ 3B 2 nδc n δ, C 3n δ B n vb 2n δ A n δc 2n δ A 2n δc n δ C n δc 2n δ 3.38 B n δc 2n δ B 2n δc n δ 3C 3 nδ B 3 nδ 3A 2 nδc n δ 3A n δb 2 nδ 3B 2 nδc n δ 6A n δb n δc n δ. Lemma 3.20. The following identities are satisfied: A n δ 1 A n δ 2B δ C δ C δ B δ B n δ A δ C δ C n δ B δ C δ A δ B δ C δ, 3.39 B n δ 1 A δ A n δ C δ B δ B δ B n δ C δ C δ C n δ A δ B δ C δ, 3.40 and where C n δ 1 A δ 2B δ C δ A n δ B δ A δ B n δ C δ B δ C δ C n δ A δ 2B δ C δ C δ B δ : B δ A δ C δ C δ B δ C δ A δ B δ C δ., 3.41 Proof. First we note that 1 δξ ξ 6 n An δ B n δξ ξ 6 C n δξ 2 ξ 5. On the other hand we obtain 1 δξ ξ 6 n 1 δξ ξ 6 n 1 δξ ξ6 A nδ B n δξ ξ 6 C n δξ 2 ξ 5 A δ B δξ ξ 6 C δξ 2 ξ 5. The final form of formulas 3.39, 3.40 and 3.41 from the following identity could be derived: a bξ ξ 6 cξ 2 ξ 5 d eξ ξ 6 fξ 2 ξ 5 α βξ ξ6 γξ 2 ξ 5, 16

where α : 1 a 2e f f e D b d f c e f d e f, β : 1 d a f e D e b f f c d e f, γ : 1 d 2e f a d 2e f f e D e d b and D : e d f f e f c f e f d e f. Lemma 3.21. The following identity for the determinant from Lemma 3.20 holds: A δ 2B δ C δ C δ B δ B δ A δ C δ C δ B δ C δ A δ B δ C δ A δ 3 B δ 3 C δ 3 3A δb δc δ A δ 2 B δ C δ 2A δ B δ 2 C δ 2 3 B δ 2 C δ 4B δ C δ 2 δ 3 2δ 2 δ 1. 3.42 Proof. By 3.16 we obtain [ 1 δξ ξ 6 1 δξ 2 ξ 5 1 δξ 3 ξ 4 ] δ 3 2δ 2 δ 1. 3.43 On the other hand, by 3.1 3.3 we get 1 δξ ξ 6 1 δξ 2 ξ 5 1 δξ 3 ξ 4 1 δξ ξ 6 1 δξ 2 ξ 5 1 δξ 3 ξ 4 A δ B δξ ξ 6 C δξ 2 ξ 5 A δ B δξ 2 ξ 5 C δξ 3 ξ 4 A δ B δξ 3 ξ 4 C δξ ξ 6 A δ 3 a B δ 3 a C δ 3 b A δb δc δ c A δ 2 B δ C δ B d A δ δ 2 C δ 2 e B δ 2 C δ f B δ C δ 2, 3.44 where see Lemma 2.3 a ξ ξ 6 ξ 2 ξ 5 ξ 3 ξ 4 1, b ξ ξ 6 ξ 2 ξ 5 ξ ξ 6 ξ 3 ξ 4 ξ 2 ξ 5 ξ 3 ξ 4 ξ ξ 6 2 ξ 2 ξ 5 2 ξ 3 ξ 4 2 3, c ξ ξ 2 ξ 3 ξ 4 ξ 5 ξ 6 1, d ξ ξ 6 ξ 2 ξ 5 ξ ξ 6 ξ 3 ξ 4 ξ 2 ξ 5 ξ 3 ξ 4 2, e ξ ξ 6 2 ξ 2 ξ 5 ξ ξ 6 ξ 3 ξ 4 2 ξ 2 ξ 5 2 ξ 3 ξ 4 3, f ξ ξ 6 ξ 2 ξ 5 2 ξ ξ 6 2 ξ 3 ξ 4 ξ 2 ξ 5 ξ 3 ξ 4 2 4. 1

Hence 3.42 follows from 3.43 and 3.44. 3.2 Reduction formulas for δ-arguments Next, our aim will be to prove some more general identities connecting the quasi-fibonacci numbers with different δ s δ 2. Let us start with the following formula: 2 δξ 2 ξ 5 δξ 3 ξ 4 n 2 δ δξ ξ 6 n 2 δ n 1 δ 2 δ n B n δ δ 2 n δ δ 2 ξ ξ6 2 δ n A n δ 2 δ ξ ξ 6 2 δ n C n δ 2 ξ 2 ξ 5. 3.45 But, we can deduce another equivalent formula n 1 2 δξ 2 ξ 5 δξ 3 ξ 4 δξ 2 ξ 5 1 δξ 3 ξ 4 n 3.46 n 1 δξ 2 ξ 5 1 δξ 3 ξ 4 n n A δ B δξ 2 ξ 5 C δξ 3 ξ 4 A n δ B n δξ 3 ξ 4 C n δξ ξ 6 n A δa n δ A δb n δξ 3 ξ 4 A δc n δξ ξ 6 A n δb δξ 2 ξ 5 B δb n δξ 5 ξ 6 ξ ξ 2 B δc n δξ 3 ξ ξ 6 ξ 4 A n δc δξ 3 ξ 4 B n δc δξ ξ 6 2 C δc n δξ 4 ξ 5 ξ 2 ξ 3 n A δa n δ A δb n δ B δc n δ A n δc δ 2B n δc δ C δc n δ n ξ ξ 6 A δb n δ A δc n δ B δb n δ A n δc δ B n δc δ C δc n δ n ξ 2 ξ 5 A δb n δ A n δb δ B δb n δ B δc n δ A n δc δ. From 3.45 and 3.46, again by linear independence of 1, ξ ξ 6 2 cos 2π and ξ2 ξ 5 2 cos 4 π over Q and by Lemma 2.8 the identities below can be generated: 18

Lemma 3.22. We have δ 2 δ n A n δ 2 and n A δa n δ A δb n δ 3.4, B δc n δ A n δc δ 2B n δc δ C δc n δ δ 2 δ n B n δ 2 n A δb n δ A δc n δ 3.48 B δb n δ A n δc δ B n δc δ C δc n δ δ 2 δ n C n δ 2 n A δb n δ 3.49. A n δb δ B δb n δ B δc n δ A n δc δ In the sequel, from the above formulas the following special identities can be obtained: the case δ 1 n A A n 1 A n A B n B C n A n C 3.50 the case δ 3 2B n C C C n, n B n 1 A B n A C n B B n A n C 3.51 C n 1 1 n A n 3 B n C C C n, n A B n A n B B B n 3.52 B C n A n C ; n A 3A n 3 A 3B n 3 3.53 B 3C n 3 A n 3C 3 2B n 3C 3 C 3C n 3, n 1 n B n 3 A 3B n 3 A 3C n 3 3.54 B 3B n 3 A n 3C 3 B n 3C 3 C 3C n 3, n 1 n C n 3 A 3B n 3 A n 3B 3 3.55 B 3B n 3 B 3C n 3 A n 3C 3. 19

The case δ 2 must be treated in another way 2 2ξ 2 ξ 5 2ξ 3 ξ 4 2n 2ξ ξ 6 2n 2 2n 2 ξ 2 ξ 5 n 2 3n 1 1 2 ξ2 ξ 5 n 2 A 3n 1 n 2 1 Bn 2 ξ 2 ξ 5 C 1 n 2 ξ 3 ξ 4 2 A 3n 1 n 2 1 Bn 2 ξ 2 ξ 5 C 1 n 2 1 ξ ξ 6 ξ 2 ξ 5 2 A 3n 1 n 2 1 Cn 2 1 Bn 2 1 Cn 2 ξ 2 ξ 5 C 1 n 2 ξ ξ 6, which, by 3.46, implies the next five identities 2 3n A n 1 2 Cn 1 2 2 2n A 2A 2n 2 A 2B 2n 2 3.56 B 2C 2n 2 A 2n 2C 2 2B 2n 2C 2 C 2C 2n 2, 2 3n B 1 n 2 1 2 2n Cn 2 A 2B 2n 2 A 2n 2B 2 3.5 B 2B 2n 2 B 2C 2n 2 A 2n 2C 2, 2 3n C 1 2 2n n 2 A 2B 2n 2 A 2C 2n 2 3.58 B 2B 2n 2 A 2n 2C 2 B 2n 2C 2 C 2C 2n, hence, after some manipulations, we obtain 2 3n A n 1 2 2 2n A 2A 2n 2 A 2C 2n 2 3.59 B 2B 2n 2 B 2C 2n 2 B 2n 2C 2, 2 3n B 1 2 2n n 2 A 2C 2n 2 A 2n 2B 2 3.60 B 2C 2n 2 B 2n 2C 2 C 2C 2n 2. We have also the identity 1 ξ ξ 6 1 δ ξ 2 ξ 5 1 ξ ξ 6 δ, i.e., 1 ξ ξ 6 n 1 δ ξ 2 ξ 5 n 1 ξ ξ 6 n, δ 20

which, by 3.1 and 3.2, yields A n B n ξ ξ 6 C n ξ 2 ξ 5 A n δ B n δ ξ 2 ξ 5 C n δ ξ 3 ξ 4 n 1 δ n ξ ξ 6 n δ n A ξ ξ 6 n n δ n B ξ 2 ξ 5 n Hence, by Lemma 2.8, the following three identities can be discovered: δ n n C. A n A n δ A n C n δ B n B n δ B n C n δ C n B n δ 3.61 n δ n A, A n C n δ B n A n δ B n C n δ C n B n δ C n C n δ 3.62 n δ n B, A n B n δ A n C n δ B n B n δ C n A n δ C n B n δ C n C n δ 3.63 n δ n C. 3.3 Polynomials associated with quasi-fibonacci numbers of order Directly from equation 3.1 we obtain for x : 1 2δ cos 2π, 1, 2, 3 x n A n δ B n δ 2 cos 2π C n δ 2 cos 4π A n δ B nδ x B nδ C nδ 2δ 2 2 cos 2 2π δ δ δ 2 1 i.e., A n δ B nδ δ A n δ B nδ δ x B nδ δ C nδ δ 2 C nδ δ 2 x 2 2x 1 2δ 2 2δ 2 1 B nδ δ W n, x;δ : δ 2 x n C n δx 2 2C n δ δb n δx x 2 C nδ x δ 2 C nδ x 2 δ, 2 δb n δ 2δ 2 1C n δ δ 2 A n δ 0 for x x, 1, 2, 3. So, by identity 3.16 the polynomial p X;δ is a divisor of the polynomial W n, X;δ for every n 3. In the sequel we obtain for δ 1 x 3 2x 2 x 1 x n C n x 2 2C n B n x B n C n A n 3.64 21

for every n N, n 3. More precisely, the following decomposition holds: hence, for δ 1, we obtain x 3 2x 2 x 1 n 2 n 2 p X;δ C 1 δx n 2 W n, X;δ, 3.65 1 B x n 2 x n C n x 2 2C n B n x B n C n A n. 3.66 1 Remar 3.23. Equation 3.65 after differentiating and some manipulating enables us to generate the sums formulas of the following form: r1 p X;δ r C δx n r r polynomial depending on W n, X;δ, p X;δ and their derivatives For example, the identity 3.65 implies the identity 3.25 for X 1. Remar 3.24. There exist other ways to obtain the relation 3.64. For example, by 3.5 we have αa n1 δ βb n1 δ γc n1 δ α β δ A n δ 2αδ β γ δ B n δ δ α 1 δγ C n δ. 3.6 We are interested when the following system of equations holds: α β δ α 2αδ β γ δ β which is equivalent to the following one: γ β 1 2 2α 2, α 3 2 2 β α β α β α δ α 1 δγ, γ 1 0. Hence, by Remar 3.11, it follows that β { α 1 2 cos 2π } : 1, 2, 3. So the equation 3.6 then has the following form: A n1 δ 2 cos 2π B n1δ 1 2 cos 2π 1Cn1 δ 1 2δ cos 2π A n δ 2 cos 2π B nδ 1 2 cos 2π 1Cn δ 1 2δ cos 2π n A 1 δ 2 cos 2π B 1δ 1 2 cos 2π 1C1 δ by 3.5 1 2δ cos 2π n1 22

or i.e., δ 2 x n1 δ x δ 1 A n δ x δ 1 x 1 B n δ δ 2 C n δ, δ 2 x n1 x 2 B n δ x 2 δb n δ δ A n δ for x : 1 2δ cos 2π, 1, 2, 3, which is equivalent to δ 1 δa n δ δ 1B n δ δ 2 C n δ 0 W n, X;δ 0. Lemma 3.25. Immediately from 3.66 we obtain the following identities: and x 3 2x 2 x 1 2 n 2 B x n 2 1 n 3 x 3 2x 2 x 1 2 n 2 B x n 3 1 n 3x n2 2n 2x n1 n 1x n nx n 1 C n x 4 2B n 2C n x 3 3A n 5B n 2C n x 2 4A n 4B n 2C n x A n 3C n, 3.68 x 3 2x 2 x 1 3 n 2 B x n 2 n 2 n 12x n4 1 4n 2 24n 32x n3 2n 2 10n 18x n2 6n 2 24n 6x n1 3n 2 9n 6x n 2n 2n 2 x n 1 n 2 nx n 2 2C n x 6 6B n 12C n x 5 12A n 24B n 18C n x 4 32A n 42B n 6C n x 3 18A n 30B n 18C n x 2 6A n 6B n 30C n x 6A n 4B N 8C n. 3.69 Corollary 3.26. We have n 2 B A n 2C n 1, 1 n 2 1 B 1 1 n A n 2B n 2C n, 1 n 3 n 2 B 2A n 2C n 1 n B n, 1 n 2 B na n B n 2nC n, 1 n 3 n 2 1 B 6 n 1 n 1 6A n 11B n 8C n, 1 23

and n 3 1 B 4 1 6A n n 11B n 8C n 1 n 2 A n 2B n 2C n B n 2. 3.4 Numerical remars about the zeros of polynomials B n x Let us set Then, B n x : B x n, n N. 1 B 1 x 1, B 2 x x 2, B 3 x x 2 2x 5, and, thus, we have the recurrence relations see 3.20 B n3 x x n2 2B n2 x B n1 x B n x, n 1. All polynomials B n x for n 2 1, N, have an even degree, for example B 5 x x 4 2x 3 5x 2 11x 25, B x x 6 2x 5 5x 4 11x 3 25x 2 56x 126. As follows from calculations, these polynomials have no real zeros for n 200. However, all polynomials B n x for n 2, N, have an odd degree, for example B 4 x x 3 2x 2 5x 11, B 6 x x 5 2x 4 5x 3 11x 2 25x 56. These polynomials have exactly one real zero, which is less than zero for n 200. The zeros s n of the consecutive polynomials B n mae a decreasing sequence from s 2 2 to s 200 2.2442546 by numerical calculations. Remar 3.2. All numerical calculations were made in Mathematica. 1 Acnowledgment The authors wish to express their gratitude to the referee for several helpful comments and suggestions concerning the first version of our paper. 1 Mathematica is registered trademar of Wolfram Research Inc. 24

Table 1: A n δ n 0 1 n 1 1 n 2 2δ 2 1 n 3 δ 3 6δ 2 1 n 4 5δ 4 4δ 3 12δ 2 1 n 5 5δ 5 25δ 4 10δ 3 20δ 2 1 n 6 14δ 6 30δ 5 5δ 4 20δ 3 30δ 2 1 n 19δ 98δ 6 105δ 5 15δ 4 35δ 3 42δ 2 1 B n δ n 0 0 n 1 δ n 2 2δ n 3 2δ 3 3δ n 4 δ 4 8δ 3 4δ n 5 5δ 5 5δ 4 20δ 3 5δ n 6 5δ 6 30δ 5 15δ 4 40δ 3 6δ n 14δ 35δ 6 105δ 5 35δ 4 0δ 3 δ C n δ n 0 0 n 1 0 n 2 δ 2 n 3 δ 3 3δ 2 n 4 3δ 4 4δ 3 6δ 2 n 5 4δ 5 15δ 4 10δ 3 10δ 2 n 6 9δ 6 24δ 5 45δ 4 20δ 3 15δ 2 n 14δ 63δ 6 84δ 5 105δ 4 35δ 3 21δ 2 25

Table 2: n A n B n C n 0 1 0 0 1 1 1 0 2 3 2 1 3 6 5 2 4 14 11 5 5 31 25 11 6 0 56 25 15 126 56 8 353 283 126 9 93 636 283 10 182 1429 636 11 4004 3211 1429 Table 3: n A n 1 B n 1 C n 1 0 1 0 0 1 1 1 0 2 3 2 1 3 8 5 4 4 22 13 13 5 61 35 39 6 10 96 113 45 266 322 8 1329 41 910 9 321 200 2561 10 10422 591 192 11 29196 16213 2015 Table 4: n A n 1 2 B n 1 2 C n 1 2 0 1 0 0 1 1 1 0 2 3 1 2 1 2 4 19 3 8 61 4 16 19 5 32 319 6 32 2069 128 335 8 128 1089 9 256 055 10 1024 22925 11 2048 4 4 16 155 32 50 64 413 32 533 256 4365 128 2835 512 184183 2048 5 8 19 16 33 16 221 64 91 16 595 64 53 512 25213 1024 8192 2048 Table 5: n A n 2 B n 2 C n 2 0 1 0 0 1 1 2 0 2 9 4 4 3 1 22 4 4 9 56 40 5 241 250 2 6 109 32 428 3169 2926 1036 8 12801 9264 4816 9 40225 34866 1312 10 152265 115316 56020 11 501489 419846 14612 26

Table 6: n A n 3 B n 3 C n 3 0 1 0 0 1 1 3 0 2 19 6 9 3 28 63 0 4 406 14 189 5 21 1365 63 6 822 3528 3969 1983 29694 2646 8 188209 83643 8390 9 43869 64820 83349 10 40820 1964361 18112 11 10530100 1419911 2336859 Table : n A n 1 2 A2 n A 2n A n 1 2 0 3 3 3 1 2 1 5 2 2 6 5 13 4 3 11 4 19 4 4 26 13 11 16 5 5 16 185 16 6 129 38 593 32 289 5 382 128 8 650 11 12389 256 9 1460 193 40169 512 10 3281 30 65169 512 11 32 639 423065 2048 References [1] D. M. Bradley, A class of series acceleration formulae for Catalan s constant, Ramanujan J. 3 1999, 149 13. [2] R. L. Graham, D. E. Knuth and O. Patashni, Concrete Mathematics. A Foundation for Computer Science, Addison-Wesley, 1994. [3] R. Grzymowsi and R. Witu la, Calculus Methods in Algebra, Part One, WPKJS, 2000 in Polish. [4] J. Kappraff and G. W. Adamson, A unified theory of proportion, Vis. Math. 5 2003, electronic. [5] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, 2001. [6] I. Niven, Irrational Numbers, MAA, 1956 appendix to the russian edition written by I. M. Yaglom. [] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, 2006. [8] P. Steinbach, Golden fields: a case for the heptagon, Math. Magazine 0 199, 22 31. [9] D. Surowsi and P. McCombs, Homogeneous polynomials and the minimal polynomial of cos2π/n, Missouri J. Math. Sci. 15 2003, 4 14. 2

[10] W. Watins and J. Zeitlin, The minimal polynomials of cos2π/n, Amer. Math. Monthly 100 1993, 41 44. 2000 Mathematics Subject Classification: Primary 11B83, 11A0; Secondary 39A10. Keywords: Fibonacci numbers, primitive roots of unity, recurrence relation. Concerned with sequences A006054, A006356, A033304, A085810, and A094648. Received January 25 2006; revised version received August 21 2006; September 5 2006. Published in Journal of Integer Sequences, September 6 2006. Revised version, correcting some errors, April 19 200. Return to Journal of Integer Sequences home page. 28