New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7

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1 Joural of Iteger Sequeces, Vol , Article New Ramauja-Type Formulas ad Quasi-Fiboacci Numbers of Order Roma Witu la ad Damia S lota Istitute of Mathematics Silesia Uiversity of Techology Kaszubsa Gliwice Polad r.witula@polsl.pl d.slota@polsl.pl Abstract We give applicatios of the quasi-fiboacci umbers of order ad the so-called sie-fiboacci umbers of order ad may other ew ids of recurret sequeces to the decompositios of some polyomials. We also preset the characteristic equatios, geeratig fuctios ad some properties of all these sequeces. Fially, some ew Ramauja-type formulas are geerated. 1

2 1 Itroductio The scope of the paper is the geeralizatio of the followig decompositios of polyomials [,, 4, 8, 9]: si π si si, 1.1 4si π 4si 4si 14, 1. 8si π 8si 8si 4 1, 1. cos π cos cos 1, 1.4 4cos π 4cos 4cos 5 61, 1.5 8cos π 8cos 8cos 4 111, 1.6 8si π cos 8si cos π 8si cos 14, 1. etc. The mai icetive for geeratig the decompositios of these polyomials is provided by the properties of the so-called quasi-fiboacci umbers of order, A δ, B δ ad C δ, N, described i [10] by meas of the relatios 1δξ ξ 6 A δb δξ ξ 6 C δξ ξ for 1,,, where ξ C is a primitive root of uity of order i.e., ξ 1 ad ξ 1, δ C, δ 0. Besides, a essetial rôle i the decompositios of polyomials discussed i the paper is played by related umbers δ C, N: A δ : A δ B δ C δ 1δξ ξ 6 1δξ ξ 5 1δξ ξ ad B δ : 1 A δ A δ 1δ cos π 1δ cos 1δ cos π 1δ cos 1δ cos 1δ cos A δ A δb δ A δc δ B δc δ B δ C δ A δ A δ A δ B δ C δ B δc δ Furthermore, to simplify the formulas, we will write A A 1, B B 1, A A 1, B B 1 ad C C 1, 1.11

3 for every N. We ote that the tables of values of these umbers ca be foud i the article [10]. Basic decompositios Witu la et al. [10] determied the followig two formulas: ad cos π cos cos B 1 A 1.1 1δ cos π 1δ cos 1δ cos A δ B δ 1 δ δ δ.. From. three special formulas follow: 1cos π 1cos 1cos A 1 6 B ,. ad si π si si A 1 B 1.4 cos π cos cos A 1 B Comparig formulas.1 ad.5 two ew idetities are geerated A B 1 ad B A 1..6 We also have the decompositio 1δξ ξ 6 1δξ ξ 5 1δξ ξ 6 1δξ ξ 4 1δ cos π 1δξ ξ 5 1δξ ξ 4 1δ cos 1δ cos π 1δ cos 6π 1δ cos 1δ cos 6π B δ 1 δ δ δ A δ 1 δ δ δ : r ;δ..

4 Now let us set Ξ : Ξ δ,ε,η δ cos π Υ : Υ δ,ε,η ε cos π Θ : Θ δ,ε,η η cos π for ay δ,ε,η C ad N 0. ε cos η cos δ cos Lemma.1 The followig geeral decompositio formula holds Ξ Υ Θ η cos,.8,.9 δ cos ε cos.10 δ εηa 1 1 δ ε δ η εη A [ δεηb δ ε η 1 δε εη ηδ A 1 δ εε η η δ 1 A 1 ]..11 Below a illustrative example coected with Lemma.1 will be preseted. Examle. A. M. Yaglom ad I. M. Yaglom [11] see also [5, problem 0] ad [6, problem 9] cosidered the followig polyomial: m1 m1 m1 wx x m x m 1 x m ad proved that it has the roots 0 π x cot, 1,,...,m. m1 I particular, for m taig ito accout that cot π cot, cot π cot we have the decompositio see also formula 6.14 below: x cot π x cot x cot x 5x x 1 ad as a corollary the decompositio x cot π x cot x cot x 5x 14x49..1 Accordig to.8 for 1, let us try to fid a liear combiatio cot π δ cos π ε cos η cos 4

5 or, the same, cos π δ cos π ε cos η cos δ cos π ε cos Decreasig powers ad taig ito accout the idetity we fid cos π δ ε η cos π cos cos π cos 1 η cos δ ε η cos δ ε η. Thus, for fidig δ, ε, η we have the liear system see Corollary.5 i [10]: δ ε η 0 δ ε η δ ε η 14 with the solutio Hece, Aalogously, we obtai cot π cot cot δ 1, ε 5, η 1. 1 cos π 1 cos π 5 cos π 5 cos 1 cos 1 cos Thus the decompositio.1 correspods to Lemma.1 with 1 cos. 5 cos,. 1 cos Ξ 1 1,5,1, Υ 1 1,5,1, Θ 1 1,5,1. Remar. The idetity.1 was foud earlier by Shevelev [4]. cos π. Remar.4 The formula.11, i some cases which are subject of our iterest, especially whe coefficiets δ, ε, η are the correspodig values of trigoometric fuctios, becomes rather complicated. Thus, i the ext two sectios we attempt to desigate the relevat coefficiets of decompositio.11, icludig the recurret coefficiets, o the grouds of ew sequeces that are easier to aalyze. 5

6 The first group of special cases of.11 Let us set a 1 [si π cos si cos π si cos ],.1 b 1 [si cos si cos π si π cos ],. c 1 [si cos si π cos π si cos ],. for 0,1,,... Lemma.1 The followig recurrece relatios hold: a 1 a b, b 1 a b c, c 1 c b,.4 for 0,1,,... ad a 0 b 0 c 0. Moreover, elemets of each sequeces {a } 0, {b } 0 ad {c } 0 satisfy the recurrece equatio x 5x 1 6x x 1 0, 0,1,,....5 i view of decompositio 1.5 a appropriate characteristic polyomial is compatible with the defiitio of umbers a, b ad c. Thefirsttwelvevaluesofumbersa a /,b b / adc c / arepreseted i Table 1. Now, let us set α [si π cos 1 si cos π 1 si cos 1 ],.6 β [si cos 1 si cos π 1 si π cos 1 ],. γ [si cos 1 si π cos π 1 si cos 1 ],.8 for 1,,... Lemma. We have α 1 0, β 1, ad γ 1. The elemets of sequeces {α } 1, {β } 1 ad {γ } 1 satisfy the system of recurrece relatios.4 ad, selectively, recurrece relatio.5. The first twelve values of umbers α α /, β β / ad γ γ / are preseted i Table 1. 6

7 Remar. There exists a simple relatioships betwee umbers α, β, γ, N, ad a, b, c, N. We have α c, β a 1 b 1, γ a 1. These relatios are easily derived from the defiitios of respective umbers, for example β 11 [ si cos cos 1 si cosπ cos π 1 si π cos cos 1 ] 11 [ si π si cos 1 si si cos π 1 si π si cos 1 ] a 1 b 1. Let us also set f 1 [cos π cos cos cos cos cos π ],.9 g 1 [cos cos cos π cos cos cos π ],.10 h 1 [ cos π 1 cos 1 cos 1 ],.11 for 0,1,,... Lemma.4 We have ad f 0 g 0 h 0 1, ad h 1 5 f 1 f g, 0, g 1 f h, 0, h 1 g h 1, 1..1 The elemets of sequeces {f } 0, {g } 0 ad {h } 0 satisfy the followig recurrece relatio see formula 1.4: Proof: By.1 we obtai ad, fially, the followig idetity x x x 1 x 0, 0,1,....1 g f 1 f,.14 h g 1 f f f 1 f,.15 f f f 1 f 1 f f 1 f1 f,

8 i.e., f f 4f 1 f f But we also have the followig decompositio of respective characteristic polyomial x 4 x 4x x x x x x 1, which implies the followig form of.16: f f f 1 f f f 1 f f 1..1 Sice f f f 1 f 0 0 so, we obtai the required idetity.1 from.1,.14 ad.15. The first twelve elemets of the sequeces {f } 0, {g } 0 ad {h } 0 are preseted i Table 1. Remar.5 We ote, that h 1 B 1 A A, 1,, is a accelerator sequece for Catala s costat see [1] ad A []. The ext lemma cotais a sequece of eight idetities ad simultaously six ewly defied auxiliary sequeces of real umbers {Ã}, { B }, { C }, {F }, {G } ad {H }. Lemma.6 The followig idetities hold 4 cos π 4 cos π cos 4 cos 4 cos π cos 4 cos 4 cos cos f g h h 1 h 4h 1 f : F,.19 cos π 4 cos π cos cos 4 cos cos cos 4 cos π cos F h f h 1 h h 1 : G,.0 cos π 4 cos cos cos 4 cos π cos cos 4 cos π cos f h h 1 F h h 1 G : H,.1 g h 1 f 1 A A 1 A G F,. f h h 1 1 A A H F,. 8

9 si π 4 cos π cos si 4 cos cos si 4 cos π cos a a h 1 h 1 h 4 1 : Ã,.4 si π 4 cos π cos si 4 cos π cos si 4 cos cos b b h 1 h 1 h 4 1 : B,.5 si π 4 cos cos si 4 cos π cos si 4 cos π cos c c h 1 h 1 h 4 1 : C,.6 Now we are ready to preset the fial result of this sectio. All recurret sequeces defied i this sectio are applied below to the descriptio of the coefficiets of certai polyomials. Theorem. The followig decompositios of polyomials hold: si π cos si cos π si cos { a B, for, α 1 B 1., for 1, si cos si cos π si π cos { b C B, for, β 1 B 1 C 1.8, for 1, si π cos π si cos si cos { c C, for, γ 1 C 1.9, for 1, cos π cos cos cos cos cos π f 1 A A 1,.0 cos cos cos π cos cos cos π g 1 A A 1 A 1,.1 9

10 cos π 4 cos π cos cos 4 cos π cos cos 4 cos cos F f h 1,. cos π 4 cos π cos cos 4 cos cos cos 4 cos π cos G f g 1,. cos π 4 cos cos cos 4 cos π cos cos 4 cos π cos H g h 1..4 Proof: Ad.0 We have hece, by 1.4 we obtai 4 cos cos 4 cos cos 1 cos π [ cos π cos cos cosπ cos cos 4 cos cos cos π cos cos π ] 4 cos cos 4 cos cos 4 cos π cos 4 cos cos Treatig, i a similar way, the followig products: 4 cos π cos 1 ad 4 cos cos 1. 4 cos π cos 1, ad addig all three received decompositios, by.1, we obtai 1 A 1 A A 1 4 cos π cos 4 cos cos 4 cos π cos 4 cos π cos 4 cos cos 4 cos π cos. But, by [10], we have hece which implies decompositio.0. A A A 1 A 0, A 1 A A 1 A A, Remar.8 It should also be oted that the followig iterestig idetity holds: 4A A A..5 10

11 4 The secod group of special cases of.11 Let us set u cos π si π cos si cos si, 4.1 v cos si π cos si cos π si, 4. w cos si π cos π si cos si, 4. x si si π si si si π si, 4.4 y si si π si π si si si, 4.5 z si π 1 si 1 si 1, 4.6 for N ad u 0 v 0 w 0 1 ad z 0 y 0 z 0, z 1 see Table, ad for sequece { w 1 / } see A [], ad see A0909 [] for sequece {z / }. The, as may by verified without difficulty, the followig recurrece relatios hold u 1 x, v 1 y z x z 1, w 1 y x, 4. x 1 u w, y 1 w v, z 1 z 1 v, for every N. Hece, we easily obtai x x w 1 x y, y y v 1 x y x z, z y z, 4.8 for every N ad fially the recurrece relatio see also equatio 1.: z 6 z 4 14z z 0, 4.9 which also satisfies the remaiig sequeces discussed i this sectio: {x }, {y }, {u }, {v } ad {w }. Remar 4.1 The characteristic polyomial of equatio 4.9 after rescalig has the form of 1. ad was recogized by Johaes Kepler The roots of this polyomial are equal to A 1 A, A 1 A ad A 1 A 4, where A 1 A...A is a regular covex heptago iscribed i the uit circle []. Theorem 4. The followig decompositios of polyomials hold: cos π si π cos si cos si u 1 u v z 1,

12 cos si π cos si cos π si v 1 v w z 1, 4.11 cos si π cos π si cos si w 1 w u z 1, 4.1 si si π si si si π si x 1 x w z 1 1, 4.1 si si π si π si si si y 1 y u z 1 1, 4.14 si π 1 si 1 si 1 z 1 z z Additioally, primarily to describe the geeratig fuctios of the sequeces {u }, {v }, {w }, {x }, {y } ad {z } see Sectio six ew recurrece sequeces were itroduced, which, from a certai poit of view, are cojugated with sequeces {u },..., {z }. Let us set u cos π 4 si si cos 4 si π si cos 4 si π si, 4.16 v cos π 4 si π si cos 4 si si cos 4 si π si, 4.1 w cos π 4 si π si cos 4 si π si cos 4 si si, 4.18 x si π 4 si π si si 4 si si si 4 si π si, 4.19 y si π 4 si π si si 4 si π si si 4 si si, 4.0 z si π 4 si si si 4 si π si si 4 si π si, 4.1 for N ad u 0 v 0 w 0 1 ad x 0 y 0 z 0 see Table 4. A easy computatio shows that the followig recurrece relatios hold u 1 u w v z 1 z 1, v 1 z 1 z 1 u, w 1 u w, 1 4.

13 for every 1. Hece, we obtai u 1 v1 w v, 4. u w1 w, 4.4 w w1 w v1 v, 4.5 w w1 w z 1 z 1 z z for every 1. Also the followig recurrece relatios hold x 1 y z, y 1 x y z, z 1 x y z, 4. for every N, which implies the idetities for every 1. y 1 x x 1 x, 4.8 x x x Theorem 4. The followig decompositios of polyomials hold: cos π 4 si si cos 4 si π si cos 4 si π si u u v, 4.0 cos π 4 si π si cos 4 si si cos 4 si π si v v w, 4.1 cos π 4 si π si cos 4 si π si cos 4 si si w u w, 4. si π 4 si π si si 4 si si si 4 si π si x w v, 4. 1

14 si π 4 si π si si 4 si π si si 4 si si y u w, 4.4 si π 4 si si si 4 si π si si 4 si π si Lemma 4.4 The followig two groups of idetities hold: z v w. 4.5 ad u z 1 v v u, v z 1 w v w, w z 1 u u w, x z 1 w w v, y z 1 u u w, z z 1 v u, u z 1 z 4 1 v u v, v z 1 z 4 1 w v w, w z 1 z 4 1 u u w, x z 1 z 4 1 w w v, y z 1 z 4 1 u u w, z z 1 z 4 1 v v u Some Ramauja-type formulas Let p,q,r R. Shevelev [4] see also [] proved that if z 1, z, z are roots of the polyomial wz z pz qz r ad z 1, z, z are all reals ad at least the followig coditio holds p r r q 0, 5.1 the the followig formula holds z1 z z p 6 r p6 r p r p 6 r 9r p r p r 5. 14

15 all radicals are determied to be real. It was verified that oly followig polyomials amog all discussed i this paper satisfy the coditio 5.1; simultaously below the respective form of idetity 5. is preseted: polyomial 1.4 ad polyomial.1 for 1 it is the classical Ramauja formula: cos π cos cos 5 ; 5. polyomial.1 for 4: cos π sec cos sec π cos sec 1 18 ; 5.4 polyomial.1 for : cos π cos cos cos cos cos π 49; 5.5 polyomial. for,, respectively: cos π 1cos cos 1cos π cos 1cos 11 49, 5.6 sec cos π sec cos secπ cos 91 ; 5. polyomial.0 for 1, polyomial.1 for 1 ad polyomial.4 for : polyomial 4.11 for 6: sec π sec sec 8 6 ; 5.8 si π cos si cos si cos π 1 61 ; polyomial 4.1 for : polyomial 4.14 for : si π si si π si si si π si si 15 si si si si π 5 ; ; 5.11

16 polyomial 4.14 for 5: si π si si π si polyomial 4.0 for : si π si si si si ; csc π cos π csc cos csc cos polyomial 4. for : 6 csc cos π csc cos cscπ cos polyomial 4. for 6: 49; 5.1 5; 5.14 csc cos π csc cos csc π cos Remar 5.1 Shevelev [4] preseted the idetities 5.10 ad 5.11 i the followig alterative form: sec π sec sec ad 1 cos π 1 cos 1 cos cos π cos π1 cos cos 1 cos cos 1 4, 5.1 respectively. Remar 5. By 5.9, 5. ad by applied the formula si α 1 cosα we obtai cos π cos π cos cos cos cos Moreover, from 5., 5.8 ad the equality 8 cos π cos cos 1 we get cos π cos cos

17 6 The sie-fiboacci umbers of order The quasi-fiboacci umbers of order itroduced by Witu la et al. i [10] ad further developed i Sectio 1 costitute the simplest, oly oe-parameter type of the so-called cosie-fiboacci umbers of order. The ame cosie-fiboacci umbers is derived from the form of the decompositio of formulas: 1δξ ξ 6 1δ cos π,...i this Sectio we shall itroduce ad aalyze a sie variety of these umbers, created i the course of decomposig formulas: 1δξ ξ 6 1iδ si π,... Theorem 6.1 Let ξ,δ C, ξ 1 ad ξ 1. The, for every N, there exist polyomials p,r,s,,l Z[δ], called here the sie-fiboacci umbers of order, so that 1δξ ξ 6 p δr δξ ξ 6 s δξ ξ 5 δξ ξ 6 l δξ ξ 5, 6.1 1δξ ξ 5 p δr δξ ξ 5 s δξ 4 ξ δξ ξ 5 l δξ 4 ξ, 6. 1δξ 4 ξ p δr δξ 4 ξ s δξ ξ 6 δξ 4 ξ l δξ ξ We have p 1 δ 1, r 1 δ δ ad s 1 δ 1 δ l 1 δ 0 see Table, where the iitial values of these polyomial are preseted. These polyomials are coected by recurrece relatios p 1 δ p δ δs δ δr δ i δl δ, r 1 δ r δδp δ, s 1 δ s δδ δδl δ, δ δ δs δ, l 1 δ l δδr δ δs δ, for N. Hece, the followig relatioships ca be deduced: δp δ r 1 δ r δ, 6.5 δs δ 1 δ δ, 6.6 δ l δ δ 1 δ 1δ δ, 6. δ r δ δ δ 1δ 1 δ1δ δ 6.8 ad, at last, the mai idetity 5 δ δ δi δ 15δ 10 δ δ 4 i δ 15δ 5 1 δ i δ 5 δ 4 i δ 5δ 1 δ This idetity satisfies, also by , the remaiig polyomials: p δ, r δ, s δ, ad l δ, N. The characteristic polyomial correspodig to idetity 6.9 has the 1

18 followig decompositio: δ i δ 15δ 10 δ 4 i δ 15δ 5 i δ 5 δ 4 i δ 5δ 1 1 δξ ξ 6 1 δξ ξ 5 1 δξ 4 ξ 1δξ ξ ξ 4 1δξ ξ 5 ξ 6 i δ i δi δ i δ 1 i δ1 δ i δ Hece, we obtai, for example, the followig explicit form of δ: δ a1δξ ξ 6 b1δξ ξ 5 where c1δξ 4 ξ d1 δξ ξ ξ 4 e1δξ ξ 5 ξ 6, 6.11 a 5ξ ξ 5 ξ ξ ξ ξ 6 5ξ ξ 4 11, b ξ ξ 5 6ξ ξ ξ ξ 6 46ξ ξ 5 5, c 46ξ ξ 5 ξ ξ ξ ξ 5 46ξ ξ 4 11, d 5ξ ξ ξ ξ ξ 5 ξ 6, e ξ ξ ξ ξ ξ 5 ξ 6. Corollary 6. The followig idetity holds: Ω δ : 1δξ ξ 6 1δξ ξ 5 1δξ 4 ξ p δ δ l δi r δs δ. 6.1 Defiitio 6. To simplify the otatio, we shall write Ω istead of Ω 1. Theorem 6.4 The followig decompositios of the polyomials hold: q ;δ : 1δi si π 1δi si 1δi si Ω δ 1 Ω δ Ω δ 1i δ i δ, 6.1 x cot π x cot x cot Moreover, the followig idetity holds: x A x Ω i x Ω δ Ω δ p δ l δ δ p δ δ p δl δ r δs δ δl δi p δr δp δs δ r δ δ s δl δ i r δl δ δs δ Ω δ4p δω δ p δ l δ δ r δs δ δl δ i r δl δ r δ δ δs δ s δl δ

19 Geeratig fuctios The geeratig fuctios of almost all sequeces discussed ad defied i this paper are preseted i this sectio. By 1.9 ad. we obtai for N: A δ 1 1δξ ξ δξ ξ δξ ξ 4 1 A δb δ 1 A δb δ 1 δ δ δ,.1 ad, i the sequel, for 1 we get A δ 1 1δξ ξ δξ ξ /;δ 1 1δξ ξ 4 1 p p 1/;δ δ δ δ 1δ δ δ 1δ δ δ,. where p,δ δ δ δ 1δ δ δ see [10]. By 1.10 ad. we get for N: B δ r 1/;δ r 1/;δ 0 B δ1 δ δ δ A δ 1 B δ1 δ δ δ A δ 1 δ δ δ,. where r,δ is defied i.. By.19 from [10] we obtai A δ 0 ξ ξ 4 1 1δξ ξ 6 ξ ξ 6 1 1δξ ξ 5 by.18 ad.1 from [10] we obtai respectively: B δ 0 ξ ξ 5 1 1δξ ξ 4 1 δ1 δ p 1/;δ ;.4 ξ ξ 6 ξ ξ 4 1 1δξ ξ 6 ξ ξ 5 ξ ξ 6 1 1δξ ξ 5 ξ ξ 4 ξ ξ 5 1 1δξ ξ 4 δδ δ p 1/;δ.5 19

20 ad C δ 0 A δ B δ A δ 0 δ p 1/;δ..6 Equally, o the grouds of.1. ad.4.6, we obtai the followig formulas: a b c à 1 B A,. 0 b a c 1 B 0 A,.8 c a b 1 B A..9 0 The special cases for 1 ca also be geerated i the followig selective ways: by.1 ad 1.5 we obtai a 0 si π 1 4 cos however, by. ad.4 we obtai ad b 0 si si 1 4 cos π 1 4 cos a1 a 1 a 0 0 si si 1 4 cos c 0 0 By 6.1 ad 6.1 we get 1 4 cos π 1 56 ;.10 si π 1 4 cos a b b Ω δ 1 1δξ ξ δξ ξ /;δ 1 1δξ ξ 4 1 q q 1/;δ Ω δ 1 Ω δ Ω δ 1 Ω δ 1 Ω δ Ω δ 1i δ i.1 δ, 0

21 ad, the special case, for 1 by 6.1 ad 6.10: Ω δ 1 1δξ ξ δξ ξ /;δ 1 1δξ 4 ξ 1 q 1 q 1 1/;δ i δi δ 1 i δi δ 1i δ i.14 δ, where q ;δ is defied i 6.1. By 1.4,.9.11 ad.19.1 the followig formulas may be obtaied: f 0 g 0 cos π 1 cos cos π 1 cos cos 1 cos cos 1 cos π 1 g h F 1 h 1 1 h 1 h 1,.15 cos 1 cos π cos 1 cos 1 f h G 1 h 1 1 h 1 h 1,.16 h cos π 1 1 cos 1 cos h 1 1 h 1 h 1 by.1 B 1 A 1 h 1 1 h 1 h 1 1 B 1 A..1 h 1 : 1. We ote that the special case of the formulas.15 ad.16 for 1 ca be treated i aother way polyomial 1.4 will be deoted here by p : h 0 ξ ξ 6 1 ξ ξ 6 ξ ξ 5 1 ξ ξ 5 ξ ξ 4 1 ξ ξ 4 p 1/ 14 p 1/ 1,.18 1

22 hece by.1: g 1 0 h1 h h 1 h 0 h 1 0 ad, fially, usig.1 agai we get f 0 g1 h g h 0 h h By 1.1, , 4.15 ad we have x si π 1 si si 1 si si π 1 si y z x 1 z 1 1 z 1 z 1,.1 0 y 0 z 0 ad z 1 : 1: z 1 0 si π 1 si si π 1 si π si 1 si si 1 si π x z y 1 z 1 1 z 1 z 1,. si 1 si si 1 si x y z 1 z 1 1 z 1 z 1,. 1 si π 1 1 si 1 1 si 1 z 1 1 z 1 z 1 1 z 1 1 z 1 z 1..4

23 By 1.1, 1.4, , 4.15, , ad 4.6, respectively, we get u 0 v 0 w 0 u 0 cos π 1 si π cos 1 si π cos 1 si π cos π 1 4 si si cos 1 4 si π si v 0 cos 1 si cos 1 si 1 v w u 1 z 1 1 z 1 z 1,.5 cos 1 si cos π 1 si 1 u w v 1 z 1 1 z 1 z 1,.6 cos π 1 si cos 1 si 1 u v w 1 z 1 1 z 1 z 1,. 1 1 cos π 1 4 si π si cos 1 4 si π si w 0 cos 1 4 si π si 1 v w u z z z 1, 1 1 cos π 1 4 si π si cos 1 4 si si cos 1 4 si si 1 u w v z z z 1, 1 1 cos 1 4 si π si 1 u v w z z z 1.

24 By 1.1, , ad 4.6, we obtai x 0 si π 1 4 si π si si 1 4 si π si y si π 1 4 si π si si 1 4 si si z si π 1 4 si si si 1 4 si π si Jorda decompositio si 1 4 si si y z x z z z 1, si 1 4 si π si x z y z z 1. 1 z 1, si 1 4 si π si x y z z z 1. 1 z 1. For sequeces A δ, B δ ad C δ we have the equality A 1 δ A δ B 1 δ Wδ B δ, N, 8.1 C 1 δ C δ where Wδ 1 δ δ δ δ 1 δ. 8. Matrix Wδ is a diagoalized matrix, ad the followig decompositio ca be obtaied 1δξ ξ Wδ A 0 1δξ ξ 5 0 A 1, δξ ξ 4 where A 1ξ ξ 6 1 1ξ ξ 5 1 1ξ ξ 4 1 ξ ξ 6 ξ ξ 6 1 ξ ξ 5 ξ ξ 5 1 ξ ξ 4 ξ ξ 4 1 ξ ξ 6 ξ ξ 5 ξ ξ 4 4

25 ad A 1 1 ξ ξ 6 ξ ξ 6 ξ ξ 6 4 ξ ξ 5 ξ ξ 5 ξ ξ 5 4 ξ ξ 4 ξ ξ 4 ξ ξ It should be oticed, that characteristic polyomial wλ of matrix Wδ is equal to 9 Tables λ 1 wλ δ p. δ. Table 1: a / b / c / α / β / γ / f g h Table : ε ω Ψ

26 Table : u v w x y z Table 4: u v w x / y/ z/

27 Table 5: F G H Ã / B / C / Table 6: Ω δ 0 1 i δ δ i δ 4i δ 1δ i δ 4 1δ 4 16i δ 4δ 4i δ 5 14i δ 5 105δ 4 40i δ 0δ 5i δ 6 0δ 6 84i δ 5 15δ 4 80i δ 105δ 6i δ 49i δ 490δ 6 94i δ 5 5δ 4 140i δ 14δ i δ

28 Table : p δ δ 1 i δ 6δ 1 4 δ 4 4i δ 1δ 1 5 5i δ 5 15δ 4 10i δ 0δ 1 6 8δ 6 0i δ 5 45δ 4 0i δ 0δ 1 1i δ 56δ 6 105i δ 5 105δ 4 5i δ 4δ 1 r δ δ δ δ δ 4 i δ 4 8δ 4δ 5 δ 5 5i δ 4 0δ 5δ 6 5i δ 6 18δ 5 15i δ 4 40δ 6δ 8δ 5i δ 6 6δ 5 5i δ 4 0δ δ s δ δ δ 0 4 4δ δ 4 5 5δ 5 10δ 10δ 4 6 i δ 6 0δ 5 0δ 10δ 6 0δ 4 18δ i δ 6 105δ 5 5δ i δ 0δ 6 0δ 4 l δ δ δ 4 δ 4 6δ 5 i δ 5 15δ 4 10δ 6 8δ 6 6i δ 5 45δ 4 15δ 6i δ 56δ 6 1i δ 5 105δ 4 1δ 10 Acowledgmets The authors wish to express their gratitude to the referee for several helpful commets ad suggestios cocerig the first versio of our paper, especially for preparig Example., which is added to the preset versio of the paper. 8

29 Refereces [1] D. M. Bradley, A class of series acceleratio formulae for Catala s costat, Ramauja J. 1999, [] R. Grzymowsi ad R. Witu la, Calculus Methods i Algebra, Part Oe, WPKJS, 000 i Polish. [] K. Irelad, M. Rose, A Classical Itroductio to Moder Number Theory, Spriger Verlag, 198. [4] V. S. Shevelev, Three Ramauja s formulas, Kvat , 5 55 i Russia. [5] D. O. Shlyarsy, N. N. Chetzov, I. M. Yaglom, The USSR Olympiad Problem Boo, W. H. Freema & Co., 196. [6] D. O. Shlyarsy, N. N. Chetsov, I. M. Yaglom, Selected Problems ad Theorems i Elemetary Mathematics, Arithmetic ad Algebra, Mir, 199. [] N. J. A. Sloae, The O-Lie Ecyclopedia of Iteger Sequeces, 00. [8] D. Surowsi ad P. McCombs, Homogeeous polyomials ad the miimal polyomial of cosπ/, Missouri J. Math. Sci , [9] W. Watis ad J. Zeitli, The miimal polyomials of cosπ/, Amer. Math. Mothly , [10] R. Witu la, D. S lota ad A. Warzyńsi, Quasi-Fiboacci umbers of the seveth order, J. Iteger Seq , Article [11] A. M. Yaglom ad I. M. Yaglom, A elemetary proof of the Wallis, Leibiz ad Euler formulas for π, Uspehi Matem. Nau VIII 195, i Russia. 000 Mathematics Subject Classificatio: Primary 11B8, 11A0; Secodary 9A10. Keywords: Ramauja s formulas, Fiboacci umbers, primitive roots of uity, recurrece relatio. Cocered with sequeces A00160, A0049, A0909, A094648, A11051, ad A Received April 19 00; revised versio received May Published i Joural of Iteger Sequeces, Jue Retur to Joural of Iteger Sequeces home page. 9