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Journal of Integer Sequences, Vol. 10 (2007),
Article 07.8.5

23 11

Quasi-Fibonacci Numbers of Order 11
Roman Witula and Damian Slota
Institute of Mathematics
Silesian University of Technology
Kaszubska 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 order 11. These numbers are defined by five conjugate recurrence equations of order
five. We study some relations and identities concerning these numbers. We present
some applications to the decomposition of some polynomials. Many of the identities
presented here are the generalizations of the identities characteristic for general recurrence sequences of order three given by Rabinowitz.

1

Introduction

Witula, Slota and Warzy´
nski [9] analyzed the relationships between the so-called quasiFibonacci numbers of order 7. In this paper we are focused on the generalization of the
identities and some facts derived in [9] to the quasi-Fibonacci numbers of order 11 An (δ),
Bn (δ), Cn (δ), Dn (δ), and En (δ), n ∈ N, δ ∈ C, defined by the relations (see also Section 3
in this paper for more details):

n
1 + δ (ξ m + ξ 10m ) = An (δ) + Bn (δ) (ξ m + ξ 10m ) + Cn (δ) (ξ 2m + ξ 9m ) +

+ Dn (δ) (ξ 3m + ξ 8m ) + En (δ) (ξ 4m + ξ 7m )
for m, n ∈ N, m ≤ 5, δ ∈ C, where ξ ∈ C is a primitive root of unity of order 11.
Additionally, the following important auxiliary sequence is considered:
An (δ) := 5An (δ) − Bn (δ) − Cn (δ) − Dn (δ) − En (δ).
1

Moreover, in order to make shorter expressions and formulas we introduce the following
notation:
m π 
m π 
i 2 π 
km := cos
,
sm := sin
,
ξ := exp
,
11
11
11

σm := ξ m + ξ −m

and

τm (δ) := 1 + δ σm

for m ∈ Z and δ ∈ C. We note that
σm = 2 k2m

and

τm (δ) = 1 + 2 δ k2m

and
σm σn = σm+n + σ|m−n| ,
σm n = σ(11−m)n ,
σm n = σ|m|n ,
1 + σ1 + σ2 + σ3 + σ4 + σ5 = 0.
Moreover we have
τm

and
τm

1
2






τm − 21 = 2 s2m ,

2
= 2 km
,

 T (k ) 
 T (k ) 
r 2s
2r s
2

= τm
= 1 + k2rs = 2 krs
,
2k2m
2k2m

(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)

where Tr (x) denote the r-th Chebyshev polynomial of the first kind. For example, from the
last formula we get
 T (k ) 
3 2m
2
= 2 k3m
,

2k2m
 T (k ) 


5 2m
4
2
2
8 k2m
− 10 k2m
+ 25 = τm
= 2 k5m
,
2k2m
2
−
τm 2 k2m

τm

3

2




= τm

(1.7)
(1.8)

for every m ∈ N.

2

Minimal polynomials, linear independence over Q

Let Ψn (x) be the minimal polynomial of cos(2π/n) for every n ∈ N. W. Watkins and
J. Zeitlin described in [8] the following identities
Y
Ts+1 (x) − Ts (x) = 2s
Ψd (x)

d|n

if n = 2s + 1 and
Ts+1 (x) − Ts−1 (x) = 2s

2

Y
d|n

Ψd (x)

if n = 2s. In the sequel if n = 2s + 1 is a prime number, then we get
Ts+1 (x) − Ts (x) = 2s Ψ1 (x)Ψn (x),

which for example implies the formula
1
Ψ11 (x) =
(T6 (x) − T5 (x))
32(x − 1)

1
(32x6 − 16x5 − 48x4 + 20x3 + 18x2 − 5x − 1)
=
32(x − 1)
1
3
3
1
= x5 + x4 − x3 − x2 + x + .
2
8
16
32
Lemma 1. Let ξ = exp(i2π/11). Then
p11 (x) =

5
Y

m=1

(x − σm ) = x5 + x4 − 4x3 − 3x2 + 3x + 1

is a minimal polynomial of the numbers σm , m = 1, 2, . . . , 5. Moreover, we have the identity
1
Ψ11 (x) = p11 (2x).
32
Corollary 2. The numbers k2m , m = 0, 1, . . . , 4 are linearly independent over Q.
Proof. If we suppose that
a + b cos(2π/11) + c cos(4π/11) + d cos(6π/11) + e cos(8π/11) = 0
for some a, b, c, d, e ∈ Q, then by the formulas

cos 2α = 2 cos2 α − 1,
cos 3α = 4 cos3 α − 3 cos α,
cos 4α = 8 cos4 α − 8 cos2 α + 1,

the number cos(2π/11) would be a root of some polynomial ∈ Q[x] having degree 6 4, which
by Lemma 1 is impossible.
Corollary 3. Every five numbers which belong to the set {1, σ1 , σ2 , σ3 , σ4 , σ5 } are linearly
independent over Q.

Proof. It follows from the identity (1.4).
The following observation will play a central role in our arguments.
Lemma 4. Let a1 , a2 , . . . , an ∈ R be linearly independent over Q and let fk , gk ∈ Q[δ],
k = 1, 2, . . . , n. If the identity holds
n
n
X
X
fk (δ) ak =
gk (δ) ak
(2.1)
k=1

k=1

for every δ ∈ Q, then fk (δ) = gk (δ) for every k = 1, 2, . . . , n and δ ∈ C.

Proof. Since fk (δ) ∈ Q and gk (δ) ∈ Q for any δ ∈ Q we get from (2.1) that fk (δ) = gk (δ) for
every k = 1, 2, . . . , n and δ ∈ Q. The last equality implies that all, respective coefficients of
polynomials fk and gk are the same. So fk (δ) = gk (δ) for all δ ∈ C and k = 1, 2, . . . , n.
3

3

Quasi-Fibonacci numbers of order 11

Let us start from the following basic result.
Lemma 5. Let δ ∈ C and n ∈ N. Then the following identities hold
n
τm
(δ) = an (δ) + bn (δ) σm + cn (δ) σ2m + dn (δ) σ3m + en (δ) σ4m + fn (δ) σ5m

(3.2m − 1)

n
τm
(δ) = An (δ) + Bn (δ) σm + Cn (δ) σ2m + Dn (δ) σ3m + En (δ) σ4m

(3.2m)

for every m = 1, 2, . . . , 5, i.e.,
τ1n (δ) =an (δ) + bn (δ) σ1 + cn (δ) σ2 + dn (δ) σ3 + en (δ) σ4 + fn (δ) σ5
=An (δ) + Bn (δ) σ1 + Cn (δ) σ2 + Dn (δ) σ3 + En (δ) σ4 ,

(3.1)
(3.2)

τ2n (δ) =an (δ) + bn (δ) σ2 + cn (δ) σ4 + dn (δ) σ5 + en (δ) σ3 + fn (δ) σ1
=An (δ) + Bn (δ) σ2 + Cn (δ) σ4 + Dn (δ) σ5 + En (δ) σ3 ,

(3.3)
(3.4)

τ3n (δ) =an (δ) + bn (δ) σ3 + cn (δ) σ5 + dn (δ) σ2 + en (δ) σ1 + fn (δ) σ4
=An (δ) + Bn (δ) σ3 + Cn (δ) σ5 + Dn (δ) σ2 + En (δ) σ1 ,

(3.5)
(3.6)

τ4n (δ) =an (δ) + bn (δ) σ4 + cn (δ) σ3 + dn (δ) σ1 + en (δ) σ5 + fn (δ) σ2
=An (δ) + Bn (δ) σ4 + Cn (δ) σ3 + Dn (δ) σ1 + En (δ) σ5 ,

(3.7)
(3.8)

τ5n (δ) =an (δ) + bn (δ) σ5 + cn (δ) σ1 + dn (δ) σ4 + en (δ) σ2 + fn (δ) σ3
=An (δ) + Bn (δ) σ5 + Cn (δ) σ1 + Dn (δ) σ4 + En (δ) σ2 ,

(3.9)
(3.10)

where
a0 (δ) = 1,

and


an+1 (δ)




b

n+1 (δ)


cn+1 (δ)
dn+1 (δ)




e (δ)


 n+1
fn+1 (δ)

b0 (δ) = . . . = f0 (δ) = 0,
=
=
=
=
=
=

an (δ) + 2δbn (δ),
δan (δ) + bn (δ) + δcn (δ),
δbn (δ) + cn (δ) + δdn (δ),
δcn (δ) + dn (δ) + δen (δ),
δdn (δ) + en (δ) + δfn (δ),
δen (δ) + (1 + δ)fn (δ)

An (δ) = an (δ) − fn (δ),

Cn (δ) = cn (δ) − fn (δ),

Bn (δ) = bn (δ) − fn (δ),

Dn (δ) = dn (δ) − fn (δ),

En (δ) = en (δ) − fn (δ),

for every n = 0, 1, 2, . . .. Hence, the following identities are derived:

An+1 (δ) = an (δ) + 2δbn (δ) − δen (δ) − (1 + δ)fn (δ)
= An (δ) + 2δBn (δ) − δEn (δ),
4

(3.11)

Bn+1 (δ) = δan (δ) + bn (δ) + δcn (δ) − δen (δ) − (1 + δ)fn (δ)
= δAn (δ) + Bn (δ) + δCn (δ) − δEn (δ),
Cn+1 (δ) = δbn (δ) + cn (δ) + δdn (δ) − δen (δ) − (1 + δ)fn (δ)
= δBn (δ) + Cn (δ) + δDn (δ) − δEn (δ),
Dn+1 (δ) = δcn (δ) + dn (δ) − (1 + δ)fn (δ)
= δCn (δ) + Dn (δ),
En+1 (δ) = δdn (δ) + en (δ) − δen (δ) − fn (δ)
= δDn (δ) + (1 − δ)En (δ),
i.e., the following system of linear equations holds

An+1 (δ) = An (δ) + 2δBn (δ) − δEn (δ),




 Bn+1 (δ) = δAn (δ) + Bn (δ) + δCn (δ) − δEn (δ),
Cn+1 (δ) = δBn (δ) + Cn (δ) + δDn (δ) − δEn (δ),


Dn+1 (δ) = δCn (δ) + Dn (δ),



En+1 (δ) = δDn (δ) + (1 − δ)En (δ),

(3.12)

where A0 (δ) = 1, B0 (δ) = C0 (δ) = D0 (δ) = E0 (δ) = 0.

Proof. Proofs of the identities (3.2m − 1) for every m = 1, 2, . . . , 5 by induction follow. For
n = 1 we have
τm (δ) = a1 (δ) + b1 (δ) σm + c1 (δ) σ2m + d1 (δ) σ3m + e1 (δ) σ4m + f1 (δ) σ5m .
Suppose that for some n ∈ N the equalities (3.2m − 1), m = 1, 2, . . . , 5, holds. Then by (1.1),
(1.2) and (3.11) we get
n+1
n
τm
(δ) = τm (δ) τm
(δ) =




= 1 + δ σm an (δ) + bn (δ) σm + cn (δ) σ2m + dn (δ) σ3m + en (δ) σ4m + fn (δ) σ5m =






= 1 + δ σm an (δ) + σm + δ (σ2m + 2) bn (δ) + σ2m + δ (σ3m + σm ) cn (δ) +






+ σ3m + δ (σ4m + σ2m ) dn (δ) + σ4m + δ (σ5m + σ3m ) en (δ) + σ5m + δ (σ5m + σ4m ) fn (δ) =






= an (δ) + 2 δ bn (δ) + δ an (δ) + bn (δ) + δ cn (δ) σm + δ bn (δ) + cn (δ) + δ dn (δ) σ2m +






+ δ cn (δ)+dn (δ)+δ en (δ) σ3m + δ dn (δ)+en (δ)+δ fn (δ) σ4m + δ en (δ)+(1+δ) fn (δ) σ5m =
= an+1 (δ) + bn+1 (δ) σm + cn+1 (δ) σ2m + dn+1 (δ) σ3m + en+1 (δ) σ4m + fn+1 (δ) σ5m ,

which by the principle of mathematical induction means that (3.2m−1) hold for every n ∈ N
and m = 1, 2, . . . , 5.
Formula (3.2m) from (3.2m − 1) and (1.4) follows for every m = 1, 2, . . . , 5.
Definition 6. We call all elements of the sequences {An (δ)}, {Bn (δ)}, . . ., {En (δ)} the
quasi-Fibonacci numbers of order (11, δ) (see Table 1 at the end of the paper). To simplify
notation we shall write {An }, {Bn }, . . ., {En } instead of {An (1)}, {Bn (1)}, . . ., {En (1)},
respectively. We call the elements of the sequences {An }, {Bn }, . . ., {En } the quasi-Fibonacci
numbers of order 11 (see Table 2 at the end of the paper).
5

Corollary 7. In the sequel, for δ = 1 we obtain the following recurrence relations:

An+1 = An + 2Bn − En ,




 Bn+1 = An + Bn + Cn − En ,
Cn+1 = Bn + Cn + Dn − En ,


D
 n+1 = Cn + Dn ,


En+1 = Dn .

(3.13)

Corollary 8. Adding identities (3.k) for odd and even k = 1, . . . , 10 respectively we obtain
the identity:
τ1n (δ) + τ2n (δ) + τ3n (δ) + τ4n (δ) + τ5n (δ) =
= 5an (δ) − bn (δ) − cn (δ) − dn (δ) − en (δ) − fn (δ)
= 5An (δ) − Bn (δ) − Cn (δ) − Dn (δ) − En (δ)
:= An (δ).

(3.14)

Hence, by (1.5) the following identities holds:
−n

2
and

An 21




=

5
X

2n
km

(3.15)

m=1

5

X
2−n An − 12 =
s2n
m.

(3.16)

m=1

Moreover, from (3.14) we obtain
5
X

m=1

coeff

n
τm
(δ); δ n




=

5
X

n
σm

n

=2

5
X

n
k2m
=

m=1

m=1






= 5 coeff An (δ); δ − coeff Bn (δ); δ n − . . . − coeff En (δ); δ n = coeff An (δ); δ n .
n

Hence, by (3.15) we get
and



coeff A2n (δ); δ 2n = 2n An

coeff A2n−1 (δ); δ

2n−1




2n−1

=2

5
X

1
2




n
(−1)m km
.

m=1

In Table 3 twelve initial values of the sequences {A(1)} (see also A062883 in [6]), {A(1/2)}
and {A(−1/2)} are presented.
Corollary 9. By (1.7) and (3.2m) we have the identity
2n
2n k3m
= An (δ) + Bn (δ) σm + Cn (δ) σ2m + Dn (δ) σ3m + En (δ) σ4m =


= An (δ) + 2 Bn (δ) k2m + Cn (δ) k4m + Dn (δ) k5m + En (δ) k3m

6

2
for δ := 2 k2m
− 23 and for every m = 1, 2, . . . , 5. Furthermore, by (1.6) and (3.2m) the
following general formula hold

2n kr2nm = An (δ) + Bn (δ) σm + Cn (δ) σ2m + Dn (δ) σ3m + En (δ) σ4m =


= An (δ) + 2 Bn (δ) k2m + Cn (δ) k4m + Dn (δ) k5m + En (δ) k3m
for δ := Tr (k2m )/(2 k2m ), m, r ∈ N. We note that x divides Tr (x) iff r is an odd positive
integer.


n
Corollary 10. Since degδ τm
= n for every m = 1, 2, . . . , 5, by (3.2m) for m = 1, 2, . . . , 5
it can be easily deduced the following formula











max deg An (δ) , deg Bn (δ) , deg Cn (δ) , deg Dn (δ) , deg En (δ) = n.

Hence, by (3.12) and Table 1, if deg(Dn (δ)) = n for n = 5, 6, . . . then also deg(Cn (δ)) = n
for n = 5, 6, . . ., and deg(En (δ)) = n for infinite many n ∈ N (see also the Section 6 in this
paper).

Corollary 11. Taking into account the equations (from the last to the first one, respectively)
of the recurrence system (3.12), we obtain:

δDn (δ) = En+1 (δ) − (1 − δ)En (δ),




δ 2 Cn (δ) = δDn+1 (δ) − δDn (δ)




= En+2 (δ) + (δ − 2)En+1 (δ) + (1 − δ)En (δ),



3

 δ Bn (δ) = δ 2 Cn+1 (δ) − δ 2 Cn (δ) − δ 3 Dn (δ) + δ 3 En (δ)
= En+3 (δ) + (δ − 3)En+2 (δ) + (3 − 2δ − δ 2 )En+1 (δ)
(3.17)

2

+(δ
+
δ
−
1)E
(δ),

n


4
3
3

δ
A
(δ)
=
δ
B
(δ)
−
δ
Bn (δ) − δ 4 Cn (δ) + δ 4 En (δ)

n
n+1



= En+4 (δ) + (δ − 4)En+3 (δ) + (−2δ 2 − 3δ + 6)En+2 (δ)



+(−δ 3 + 4δ 2 + 3δ − 4)En+1 (δ) + (δ 4 + δ 3 − 2δ 2 − δ + 1)En (δ)
and, finally:

δ 4 An+1 (δ) − δ 4 An (δ) − 2δ 5 Bn (δ) + δ 5 En (δ) = 0,

(3.18)

i.e.
En+5 (δ) + (δ − 5)En+4 (δ) + (−4δ 2 − 4δ + 10)En+3 (δ)+
+ (−3δ 3 + 12δ 2 + 6δ − 10)En+2 (δ) + (3δ 4 + 6δ 3 − 12δ 2 − 4δ + 5)En+1 (δ)+
+ (δ 5 − 3δ 4 − 3δ 3 + 4δ 2 + δ − 1)En (δ) = 0. (3.19)
Immediately from equations (3.13) or (3.17) and (3.19) for δ = 1 we obtain the following
formulas:

En+1 = Dn ,



Cn = Dn+1 − Dn ,
(3.20)
Bn = Dn+2 − 2Dn+1 + Dn−1 ,



An = Dn+3 − 3Dn+2 + Dn+1 + 2Dn ,
7

and
Dn+4 − 4Dn+3 + 2Dn+2 + 5Dn+1 − 2Dn − Dn−1 = 0.

(3.21)

The characteristic polynomial p11 (X; δ) of the recurrence equation (3.19) has the following
decomposition (see general formula (3.39) below):
p11 (X; δ) := X5 + (δ − 5) X4 + (−4δ 2 − 4δ + 10) X3 + (−3δ 3 + 12δ 2 + 6δ − 10) X2 +
+ (3δ 4 + 6δ 3 − 12δ 2 − 4δ + 5) X + (δ 5 − 3δ 4 − 3δ 3 + 4δ 2 + δ − 1) =
=

5
Y

m=1



X − τm (δ) . (3.22)

Sketch of the proof: We have
5
X

τm (δ) =

m=1

1
τm (δ) τn (δ) =
2
1≤m 2, have an even degree, for example:
D7 (x) = x4 + 4x3 + 14x2 + 43x + 126,
D9 (x) = x6 + 4x5 + 14x4 + 43x3 + 126x2 + 357x + 993.

As follows from calculation, these polynomials have no real roots for n 6 200.
However, all polynomials Dn (x) for n = 2k, k ∈ N, k > 2, have an odd degree, for
example:
D6 (x) = x3 + 4x2 + 14x + 43,
D8 (x) = x5 + 4x4 + 14x3 + 43x2 + 126x + 357.
Remark 24. Polynomials D2k (x) have exactly one real root sk , which is less than zero (for
k 6 100). The roots sk form an increasing sequence from s2 = −4 to s100 = −2.699746 (by
numerical calculation). All numerical calculations were performed using Mathematica.1

10

Jordan decomposition

For sequences An (δ), Bn (δ), Cn (δ), Dn (δ) and En (δ) we have the identity:




An+1 (δ)
An (δ)
 Bn+1 (δ) 
 Bn (δ) 




 Cn+1 (δ)  = W(δ)  Cn (δ)  ,
n ∈ N,




 Dn+1 (δ) 
 Dn (δ) 
En+1 (δ)
En (δ)
1

Mathematica is registered trademark of Wolfram Research Inc.

23

(10.1)

where





W(δ) = 



1 2δ
δ 1
0 δ
0 0
0 0

0
δ
1
δ
0

0 −δ
0 −δ
δ −δ
1
0
δ 1−δ





.



(10.2)

Matrix W(δ) is diagonalizable, because of the following decomposition:


W(δ) = A · diag τ1 (δ), τ2 (δ), . . . , τ5 (δ) · A−1 ,

where:

(10.3)



A = K1 , K2 , K3 , K4 , K5

and

 −4

−3
−2
−1
−2
−1
−1
−2
−3
−2
−3 T
Km = σm
+ 2 σm
− 2 σm
− 2 σm
, σm
− σm
− 1, −σm
− σm
, −σm
− σm
, −σm
,

for every m = 1, 2, . . . , 5. We note, that:



A−1

11 −3 −20


 −2
2
 
1 

7
=
L1 , L2 , L3 , L4 , L5 ·  −13

11

1 −1

3 −2

where

8

16




−7 −11 


22 −15 −17  ,

−3
3
4 

−5
4
4
7


T
Lm = σ1m+3 , σ2m+3 , σ3m+3 , σ4m+3 , σ5m+3 ,

for m = 1, 2, . . . , 5.
Notice, that the characteristic polynomial w(λ) of the matrix W(δ) has the form:
 
w(λ) = p11 λ; δ .

11

Final remarks

Quasi-Fibonacci numbers of order (7, δ) and (11, δ) constitute a special case of the so-called
quasi-Fibonacci numbers of order (k, δ), where k is an odd positive integer, k ≥ 5 and δ ∈ C.
These numbers are defined to be the elements of the following sequences of polynomials:



an,i (δ) n∈N ⊂ Z[δ],

i = 1, 2, . . . ,

1
ϕ(k),
2

where ϕ is the Euler’s totient function, which are determined by the following relations:
X

n
an,f (i) (δ) (ξ i l + ξ i (k−l) )
(11.1)
1 + δ (ξ l + ξ k−l ) = an,1 (δ) +
i∈{2,3,...,(k−1)/2}
(i,k)=1

24

for l = 1, 2, . . . , k−1
, (l, k) = 1, n ∈ N, where ξ ∈ C is a primitive root of unity of order k
2
and where f is the increasing bijection of the set {i ∈ N : 2 ≤ i ≤ (k − 1)/2 and (i, k) = 1}
onto the set {2, 3, . . . , ϕ(k)/2}. In the sequel, if k is a prime number, relations (11.1) have
the following simpler form:
l

1 + δ (ξ + ξ

k−l

(k−3)/2
X

n
an,i+1 (δ) (ξ i l + ξ i (k−l) )
) = an,1 (δ) +

(11.2)

i=1

for l = 1, 2, . . . , k−1
and n ∈ N.
2
Quasi-Fibonacci numbers of order (k, δ) possess many interesting properties and natural
applications. We remark that these are for the most cases derived immediately from relations (11.1). For example the quasi-Fibonacci numbers enable us to describe the coefficients
of the following polynomials in the pure algebraic language:
Y 
Y 






lπ
lπ
lπ
X − cosn 22k+1
X − sin 22k+1
,
,
cosn 22k+1
l∈{1,2,...,k}
(l,2k+1)=1

l∈{1,2,...,k}
(l,2k+1)=1

etc. (see for example [9, 10]). It is possible also to do it for many other trigonometric sums,
for example, in [10] the following identity was derived (plus twelve others like this one):
q
q
q
q
√
3
3
2π 3
4π
8π
2π
4π 3
8π 3
2 cos( 7 ) 2 cos( 7 ) + 2 cos( 7 ) 2 cos( 7 ) + 2 cos( 7 ) 2 cos( 7 ) = −2 − 3 49;
which is a variation of the known identity of Ramanujan (see [1]):
q
q
√

q

q




3
3
cos 27π + 3 cos 47π + 3 cos 87π = 3 12 5 − 3 7 .
It seems almost definite that some types of the identities will be characteristic of only a
certain kind of the quasi-Fibonacci numbers of order (k, δ), with respect to the odd k ∈ N,
k ≥ 5. Accordingly, it is relevant to analyze these numbers separately and independently,
for different values of k.
We note that quasi-Fibonacci numbers of order (5, 1) are equal to the classical Fibonacci
numbers. Moreover, for simplicity of notation, the quasi-Fibonacci numbers of order (11, δ)
discussed in our paper are denoted by (see the relations (11.2) above for k = 11 and the
relations (3.2m) for m = 1, 2, 3, 4, 5 in Section 3):
An (δ) := an,1 (δ)
Dn (δ) := an,4 (δ)

12

Bn (δ) := an,2 (δ)
En (δ) := an,5 (δ).

Cn (δ) := an,3 (δ)

Acknowledgements

The authors wish to thank the reviewer for his valuable criticisms and suggestions, leading
to the present improved version of our paper.

25

Table 1:
n
0
1
2
3
4
5
6
7
n
0
1
2
3
4
5
6
7
n
0
1
2
3
4
5
6
7

An (δ) = A11,n (δ)
1
1
2δ 2 + 1
6δ 2 + 1
6δ 4 + 12δ 2 + 1
−δ 5 + 30δ 4 + 20δ 2 + 1
19δ 6 − 6δ 5 + 90δ 4 + 30δ 2 + 1
−7δ 7 + 133δ 6 − 21δ 5 + 210δ 4 + 42δ 2 + 1
Bn (δ) = B11,n (δ)
0
δ
2δ
3δ 3 + 3δ
12δ 3 + 4δ
9δ 5 + 30δ 3 + 5δ
−δ 6 + 54δ 5 + 60δ 3 + 6δ
28δ 7 − 7δ 6 + 189δ 5 + 105δ 3 + 7δ
Cn (δ) = C11,n (δ)
0
0
δ2
3δ 2
4δ 4 + 6δ 2
−δ 5 + 20δ 4 + 10δ 2
14δ 6 − 6δ 5 + 60δ 4 + 15δ 2
−7δ 7 + 98δ 6 − 21δ 5 + 140δ 4 + 21δ 2

26

Dn (δ) = D11,n (δ)
0
0
0
δ3
4δ 3
4δ 5 + 10δ 3
−δ 6 + 24δ 5 + 20δ 3
14δ 7 − 7δ 6 + 84δ 5 + 35δ 3
En (δ) = E11,n (δ)
0
0
0
0
δ4
−δ 5 + 5δ 4
5δ 6 − 6δ 5 + 15δ 4
−6δ 7 + 35δ 6 − 21δ 5 + 35δ 4

Table 2:
n
0
1
2
3
4
5
6
7
8
9
10
11

An
1
1
3
7
19
50
134
358
959
2569
6886
18461

Bn
0
1
2
6
16
44
119
322
868
2337
6284
16885

Cn
0
0
1
3
10
29
83
231
636
1735
4708
12727

Table 3:
Dn
0
0
0
1
4
14
43
126
357
993
2728
7436

En
0
0
0
0
1
4
14
43
126
357
993
2728

n
0
1
2
3
4
5
6
7
8
9
10
11

An (1)
5
4
12
25
64
159
411
1068
2808
7423
19717
52529

An (1/2)
5
9/2
25/4
39/4
257/16
437/16
1517/32
2671/32
38017/256
68169/256
245935/512
1782735/2048

An (−1/2)
5
11/2
33/4
55/4
385/16
693/16
2541/32
4719/32
70785/256
133705/256
508079/512
3879865/2048

Table 4:
n
0
1
2
3
4
5
6
7
8
9
10
11

An (1/2)
1
1
3/2
5/2
35/8
251/32
911/64
3327/128
6095/128
5593/64
164407/1024
604487/2048

Bn (1/2)
0
1/2
1
15/8
7/2
209/32
779/64
1449/64
1345/32
19941/256
36903/256
545721/2048

Cn (1/2)
0
0
1/4
3/4
7/4
119/32
241/32
1897/128
229/8
27949/512
105641/1024
12397/64

27

Dn (1/2)
0
0
0
1/8
1/2
11/8
207/64
7
3689/256
7353/256
57361/1024
220363/2048

En (1/2)
0
0
0
0
1/16
9/32
53/64
65/32
289/64
4845/512
19551/1024
4807/128

Table 5:
n
0
1
2
3
4
5
6
7
8
9
10
11

An (−1/2)
1
1
3/2
5/2
35/8
253/32
935/64
3509/128
6655/128
3179/32
195415/1024
753709/2048

Bn (−1/2)
0
-1/2
-1
-15/8
-7/2
-209/32
-781/64
-1463/64
-1375/32
-20757/256
-39325/256
-598345/2048

Cn (−1/2)
0
0
1/4
3/4
7/4
121/32
253/32
2079/128
33
34067/512
136609/1024
136367/512

Dn (−1/2)
0
0
0
-1/8
-1/2
-11/8
-209/64
-231/32
-3927/256
-8151/256
-66671/1024
-269951/2048

En (−1/2)
0
0
0
0
1/16
11/32
77/64
55/16
561/64
10659/512
48279/1024
52877/512

References
[1] R. Grzymkowski and R. Witula, Calculus Methods in Algebra, Part One, WPKJS, 2000
(in Polish).
[2] A. Mostowski and M. Stark, The Elements of Higher Algebra, PWN, 1974 (in Polish).
[3] S. Paszkowski, Numerical Applications of Chebyshef Polynomials and Series, PWN,
1975 (in Polish).
[4] S. Rabinowitz, Algorithmic manipulation of third-order linear recurrences, Fib. Quart.
34 (1996), 447–464.
[5] T. J. Rivlin, Chebyshef Polynomials from Approximation Theory to Algebra and Number
Theory, 2nd ed., Wiley, 1990.
[6] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, 2007.
[7] D. Surowski and P. McCombs, Homogeneous polynomials and the minimal polynomial
of cos(2π/n), Missouri J. Math. Sci. 15 (2003), 4–14.
[8] W. Watkins and J. Zeitlin, The minimal polynomials of cos(2π/n), Amer. Math. Monthly
100 (1993), 471–474.
[9] R. Witula, D. Slota and A. Warzy´
nski, Quasi-Fibonacci numbers of the seventh order,
J. Integer Seq. 9 (2006), Article 06.4.3.
[10] R. Witula and D. Slota, New Ramanujan-type formulas and quasi-Fibonacci numbers of order 7,
J. Integer Seq. 10 (2007), Article 07.5.6.

28

2000 Mathematics Subject Classification: Primary 11B83, 11A07; Secondary 12D10, 39A10.
Keywords: Fibonacci numbers, primitive roots of unity, recurrence relation.
(Concerned with sequence A062883.)

Received December 9 2006. Revised version received August 3 2007. Published in Journal
of Integer Sequences, August 5 2007.
Return to Journal of Integer Sequences home page.

29