COUPLED RECURRENCE RELATIONS
The 2-Fibonacci Sequences
2.3 The Scheme (2.2)
We shall study next the properties of the sequences for the scheme (2.2) and will conclude with a second basic theorem, as above. Since the proofs of the results below are similar to those above, we shall only list the results.
The first ten terms of the sequences defined in (2.2) are:
n 0 1 2 3 4 5 6 7 8 9
an a c b + c b + c + d a + b + c + 2.d 2.a + b + 2.c + 3.d 3.a + 2.b + 4.c + 4.d 4.a + 4.6 + 7.c + 6.d 6.a + 7 . 6 + l l . c + 1 0 . d 1 0 . a + 1 1 . 6 + 1 7 . c + 1 7 . d
Pn b d a + d a + c + d a + b + 2.c + d a + 3.b + 3.c + 2.d 2.a + 3.6 + 4.c + 4.d 4.a + 4.6 + 6 . c + 7 . d 7.a + 6.b+10.c+ll.d l l . o + 1 0 . 6 + 1 7 . c + 1 7 . d
Theorem 2.12: For every integer n > 0:
(a) a6.n + /30 = As.n + a0, (b) a6.n + 1 + fa = Pe.n+i + ati,
(c) a6.n+2 +a0+ Pi = Ps.n+2 + /Jo + <*1.
(d) a6.n + 3 + a0 = /?6.n+3 + A)>
(e) a6.n+ 4 +OLi= P%.n+A + Pi,
( / ) a6. „ + 5 + C*i + /J0 = Pe.n+5 + Pi +a0-
Theorem 2.13: For every integer n > 0:
n
(a) an+2 = £ Pi + ai,
i=0
(b)p
n+2= Y: <*i+0i-
i=0
Theorem 2.14: For the integer k > 0:
6.fc
(a) J2 (an - Pi) = a0 - ft.,
i=0 6.fc+l
(b) £ (ai - pi) = a0 - Po + ai ~ fa,
6.k+2
(c) £ fa - Pi) = 2 . ( a1- f t ) .
i=0 6.fc+3
(d) £ (a i_/ 3 i) = -a o + /?2 + 2. ( a1- f t ) ,
i=0 6.k+4
(e) X) ( a * - f t ) = - a0 + f t > + a i - f t ,
i=0 6.fc+5
(/) £ K-ft) = o.
Theorem 2.15: For every integer n > 0:
an+2 + fti+2 = -Fn+i-(ao + Po) + Fn+2.(ai + ft).
As above, we express the members of the sequences { a ; } ~0 and {ft}g.0, when n > 0 by (2.5).
It is interesting to note that the Theorems 2.6, 2.8 and 2.9 with identical forms are valid here.
Let ip be the integer function defined for every k > 0 by:
r 0 1 2 3 4 5
V»(6. k + r) 1 0 - 1 - 1 0 1
Obviously, for every n > 0, tp(n + 3) = —ip{n).
Using the definition of the function if>, the following are easily proved by induction:
T h e o r e m 2.16: For every integer n > 0:
(a)11n=S1n + rP(n),
(c)ri = 63n+iP(n + 4),
T h e o r e m 2.17: For every integer n > 0:
(a) (b) (c) (d)
(e) (/)
7^+2
< - /2 „
)n+2
-yi
'Y3
7n+2
-V/n+2 4 73
In
-
=
=
=
=
=
7 ^ + 1 + 7 ^ + ^ + 3), 7n+l+7*+V'(™)>
T £ + IK>*)>
73J+1+7n+<Kn + 4),
7 * + i + 7 ^ + ^ + 1),
74 + V ( n + 4).
From the above theorems, we obtain the equations:
ll=6n=1-.{Fn^+^{n + Z)),
<Y3n=6n = l.(Fn-.1+iP(n + 4)),
We may now state our second basic theorem.
S E C O N D B A S I C T H E O R E M . If n > 0, then a „ = | . ( ( Fn_ i + ^ ( n ) ) . a + ( Fn_1V ( n + 3).6
+ (Fn + i/>(n + 4)).c + (Fn + i/>(n + l)).d)
= | . ( ( o + 6).Fn_i + (c + d).Fn
+ r/>(n).a + i/>(n + 3).6 + i/)(n + 4).c + ^ ( n + 1), /?„ = | . ( ( Fn_ i + ^ ( n + 3)).a + (Fn_i^(n).&
+ (Fn + ^(n + l)).c + (Fn + </>(n + 4)).d)
= | . ( ( a + fc).Fn_i + (c + d).Fn
+ V(n + 3).o + </>(n).& + ^ ( ^ + l)-c + <Kn + 4) -
The sequences (2.4) are actually two independent Fibonacci sequences of the form {-Fi(a,c)}?^0 and {Fi(b,d)}^0. It is easily seen that the se- quences (2.3) can be expressed through the sequences {Fi(a,d)}^l0 and {Fi(6,c)}g0, for n > 1, thus:
CH2.n = F2.n{a,d),
Q2.71+1 = - p 2 .n +i ( 6 , c ) ,
l#2.n = F2.n(b,c),
fa.n+l = F2.n+l{a,d),
On the basis of what has been done above, one could be led to generalize
and examine sequences of the following types:
a0
O-n+2
Pn+2
Oto O-n+2
Pn+2
=
=
=
=
=
=
a, fa = 6, a i = c, pi p.Pn+i + q.p„, n > 0 r.an+i + s.an, n>0 a, p0 = 6, a i = c, Pi p . an+ i + q.pn, n > 0 r./9„+i + s.an, n > 0 for the fixed real numbers p, q, r, and s.
This problem was formulated in 1986, but up to this moment it is open.
Finally, we shall describe one more modification of the above schemes, and outline the solutions. The new schemes have the following forms:
I type (trivial):
a0 = a, po — b, Qi = c, Pi = d an+2 = «n+i -a», n > 0 Pn+2 = Pn+1 ~ Pn, Tl > 0 II type (trivial):
a0 = a, P0 = b, ai=c, Pi = d an+2 = Pn+i - <xn, n > 0 Pn+2 = Q-n+1 - Pn, « > 0 III type:
Q0 = o, Po = b, Qi = c, ft = d
a n + 2 = « n + l - A i , ^ > 0 A i + 2 = Pn+1 - " „ , 71 > 0
IV type:
Qo = a, /30 = &, a i = c, P\ = d
OLn+2 = Pn+1 ~ Pn, n>0
Pn+2 = an+i -an, n > 0
For the first case (I type) we obtain directly, that:
Pn
a, c, c — a, -a, -c, a — c,
b, d, d-b, -b, -d, b-d,
if n = 0 (mod 6) if n = 1 (mod 6) if n = 2 (mod 6) if n = 3 (mod 6) if n = 4 (mod 6) if n = 5 (mod 6)
if n = 0 (mod 6) if n = 1 (mod 6) if n = 2 (mod 6) if n = 3 (mod 6) if n = 4 (mod 6) if n = 5 (mod 6)
and for the second case (II type), we find that:
an = <
A.
a, c, d — a, -6, -d, b — c,
b, d, c — b, -a, -c, a — d,
if n = 0 (mod 6) if n = 1 (mod 6) if n = 2 (mod 6) if n = 3 (mod 6) if n = 4 (mod 6) if n = 5 (mod 6)
if n = 0 (mod 6) if n = 1 (mod 6) if n = 2 (mod 6) if n = 3 (mod 6) if n = 4 (mod 6) if n = 5 (mod 6)
When the other two cases are valid, the following two theorems can be proved analagously to the respective ones given above for the first schemes.
T h e o r e m 2.18: For every integer n > 0, for the third scheme:
an = | . ( ( Fn_ i + V ( n ) ) . a - ( Fn_ i + t f ( n + 3)).6
+ (Fn + rl>(n + 4)).c - (Fn + i/,(n + l)).d), Pn = | . ( - ( K _1+ V ( n + 3)).a + ( Fn_ i + V ( n ) . 6
- {Fn + V<n + l)).c + (Fn + V(n + 4)).d).
T h e o r e m 2.19: For every integer n > 0, for the fourth scheme:
an = i . ( ( - l ) » . ( Fn_1+ 3 . [ 2 ± a ] - n + l).a
+ ( - l ) n + i . ( Fn_1- 3 . [ 2 ± 2 ] + n - l ) . 6 + ( - l ) ^1. ^ - 3.[a=Z] + n - 3).c + ( - l )n. ( Fn- 3 . [ s ^ ] - n + 3).d), /?„ = |.((_i)»»+i.(irn_1_3.[2] + n_ i ) .a
+ ( - l ) » . ( Fn_1+ 3 . [ 2 ] - n - l ) . 6 + ( - l )n. ( Fn + 3 . [ = = * ] - n + 3).c + ( - l ) "+ 1. ( Fn - 3 . [ ^ ] + n - 3).d).
Extensions of t h e Concepts of 2-Fibonacci Sequences
A new direction for generalizing the Fibonacci sequence was described in Chapter 2. Here, we shall continue that direction of research.
Let CI,C2,.--,CQ be fixed real numbers. Using C\ to C&, we shall construct new schemes which are of the Fibonacci type and called 3-F- sequences. Let for n > 0
do = Ci, &o = Cii Co = C3, a\ = C4, 61 = C5, c\ = CQ an+2 = xl+1 + yln
bn+2 = X2n+i + Vl
Cn+2 = X3n+i + Vl
where < xn+1,xn+l,x^+l > is any permutation of < an+i, bn+1, cn+i >
and < xn,xn,x^ > is any permutation of < an,bn,cn > . The number of different schemes is obviously 36.
In [22], the specific scheme (n > 0)
do = Ci> bo = C2, Co = C3, a\ = d, b\ = C5, ci = C6
fln+2 — frn+1 "+" cn
bn+2 — Cn+1 + On
29
Cn+2 — On+1 + bn
is discussed in detail. For the sake of brevity, we devise the following representation for this scheme:
(
a b c \ b c a \ (3.1)c a b )
Note that we have merely eliminated the subscripts and the equal and plus symbols so that our notation is similar to that used in representing a system of linear equations in matrix form. Using this notation, it is important to remember that the elements in their first column are always in the same order while the elements in the order column can be permuted within that column. Every elememt a, b and c must be used in each column.
We now define an operation called substitution over these 3-F-sequences and adopt the notation \p,q]S, where p,q £ {a,b,c}, p ^ q. Applying the operation to S merely interchanges all occurrences of p and q in each column.
For example, using (3.1), we have
(
c b a \ b a c I (3.2)a c b J
Note that in the result we do not maintain the order of the elements in the first column. To maintain this order we interchange the first and last rows of (3.2) to obtain:
(
b a c (3.3) a c b \c b a ) which corresponds to the scheme (n > 0)
do = Clt bo = C2, Co = C3, ai = C4, &i = C5, C1 = C6
an+2 = c„+i + bn
bn+2 — an+i + cn
cn+2 — bn+i + an
where C[, C2,..., C'G are real numbers.
We shall say that the two schemes S and 5" are equivalent under the operation of substitution and denote this by S <-> S'.
It is now obvious that for any two 3-F-sequences S and S", if \p, q]S «->
5 ' , then \p,q]S' <-> S. To investigate the concept of equivalence to a deeper extent, it is necessary to list all 36 schemes:
S2 = b b c S3 =
a a b b b a
a a b
Sg = I b C C | Sg = c b a
a a c
Su = | b c a | Si 2 = e b b
a b a
S14 = [ b a c | Si 5 = c c b
a b b
Sn = | b a a | Si 8 = c c c
a 6 6 \
S20 = I 6 c c S2i = c a a I
a b c a b c a b c a b c a b c a b c a b c
a c b a b c a b c a c b b c a b a c b a c
a b c b c a c a b c b a a b c b c a c a b
S22 = [ b a b \ S23 = I b c a \ S24 =
S25 = \ b a b ] S26 = [ b a c S27 =
5-28 = I b b c I 52g = I b a a \ S30 =
S31 = \ b b a \ S32 = I 0 v c I 033
S34 = I 6 a 6 S35 = I 6 6 a 53 6 =
\ c b a / \ c a b ) Note that S = S23 and S' = 530, so that 523 <-* ^ o •
We say that a 3-F-sequence S is trivial if at least one of the resulting sequences is a Fibonacci sequence. Otherwise, 5 is said to be an essential generalization of the Fibonacci sequence.
Observe that there are ten trivial 3-F-sequences. They are Si, S2, S3, S4, S5, S10, S13, Sir > S27 and 536-
These 10 schemes are easy to detect since they have at least one row all with the same letter. Furthermore; for these schemes one of the three possible substitutions returns the scheme itself. For example,
[b,c]Si <-> Si, i = 1 , 2 , 3 , 4
[a,b]Si^Si, i = l,5,13,17 [a,c]Si <-> Si, i = 1,10,27,36.
The twenty-six remaining schemes are essential generalizations of the Fi- bonacci sequence. For eight of these schemes, the result is independent of the substitution made. That is,
\p,q]S6 <-> S9
fa, q]Sl5 <-> 525
fa, q]S20 <-> S3 3
fa, q]Sl3 ^ 530
for all p,q € {a,b,c}. This means the substitution operation for these schemes is cyclic of length 2.
For the other eighteen essential generalizations of the Fibonacci se- quence schemes, all three possible substitutions generate three different schemes. For example,
[b, c]S7 <-> Si2 [b, c]Ss <-> S u [6, c]Si6 <-> S26
[a,c]S7 «-> Su [a,c]S8 <-> S2i [a,c]Si6 <-> S29
[a, b]S7 «-+ S31 [a, b]S8 «-> S35 [a, 6]Si6 <-> S34
All of the substitutions associated with the remaining eighteen schemes and their results are conveniently illustrated by the following three figures.
That is, these pictures determine all possible cycles.
Figure 1
Figure 2
Figure 3
For example,
[a,b]S29 <-> Si9 and
[a,6]([fe,c]([a,c]S24))^S29.
Note that the figures tell us that many of the schemes are independent.
That is, Sig and S24 are independent. In fact, Sis is related only to the six schemes listed in Fig. 3. Similar results can be found for the other schemes.
The closed form equation of the members for all three sequences of the scheme S23 is given in [22].
For every one of the above groups we shall give the recurrence relations of its members.
Let everywhere
a® — Ci, bo = C*2, Co = C3, 01 = 6*4, bi — C§, ci = C$
and n > 0 be a natural number, where C\,CI,---,CG are given constants and lx' is one of the symbols 'a', '6' and 'c'.
Group / contains the schemes S& and Sg, where:
{
On+2 = O-n+1 + bnbn+2 = bn+l + C„
Cn+2 = Cn+ i + an
The recurrence relation for this scheme is:
•^n+6 = o.Xn+§ o.Xn_)_4 + ^ n + 3 r ^ n j i.e.
fln+6 = 3 . an+ 5 — 3.on_)-4 + a „ + 3 + an,
bn+6 — 3.fe„+5 — 3.6„+4 + bn+3 + bn,
Cn+6 ~ 3.C„+5 — 3.C„+4 + Cn_f-3 + C„.
Group II contains the schemes S15 and 525 , where:
(
O-n+2 = &n+l + « n bn+2 = C„+i + bn C J I + 2 = O-n+1 + cnThe recurrence relation for this scheme is:
Xn+6 = 3.2„+ 4 + Xn+3 - 3.Xn+2 + Xn. Group III contains the schemes S20 and 533 , where:
{
a-n+2 = bbn+2 = Cn+l + Cn n+i + bn Cn+2 — O-n+1 + anThe recurrence relation for this scheme is:
Xn+6 = xn+3 + 3.Z„+2 + 3.£n +i + Xn. Group IV contains the schemes 523 and 533 , where:
{
O-n+2 = bbn+2 = C„+l + an+l + Cn nCn+2 = O.n+1 + bn
The recurrence relation for this scheme is:
Xn+6 — ^-Xn+3 ~r Xn-
Group V contains the schemes Sj, S12, S14, S22, S28 and 531, where:
(
an+2 = O-n+l + bnbn+2 = cn+i + an
Cn+2 = bn+l + Cn The recurrence relation for this scheme is:
Xn+6 = Xn+h + 2.Z„+4 - 2.Xn+3 + Xn+2 ~ Xn.
Group VI contains the schemes Sg,Sn, Sis, 821,832 and 535, where:
[ O-n+2 — O-n+1 + bn
58 : s bn+2 = cn+i + cn cn+2 = bn+i + an
The recurrence relation for this scheme is:
Xn+6 = Xn+5 + Xn+4 - Xn+2 + Xn+i + Xn.
Group VII contains the schemes S^, S\g, S24, S26, S29 and 534, where:
{
O-n+2 = bbn+2 = C „ + l + Cn+\ + 8 n nCn+2 = an+l + bn
The recurrence relation for this scheme is:
Xn+6 = Xn+4 + 1.Xn+3 + 2 . X „+ 2 - Xn+i - Xn.
Following the above idea and the idea from [22], we can construct 8 different schemes of generalized Tribonacci sequences in the case of two sequences.
We introduce their recurrence relations below.
Let lx' be one of the symbols 'a' and '&'.
The different schemes are as following:
n + l 1 (In +1 + bn
rp . f on +3 — an+2 + ar
\ bn+3 = bn+2 + bn
rp . \ O-n+3 = O-n+2 + an+l + bn
I bn+3 — bn+2 + bn+i + an
[3 : \ On+3 — « n + 2 + bn+l + « n
I bn+z = bbnn +2 + a « + i + bn
n+2 + bn+l + bn
'n+2 + O-n+1 + O-n rp . f On+3 — &
\ bn+3 = b,
rp . \ O-n+3 = bn+2 + 0,n+l + dn
I bn+3 = an+2 + bn+i + bn
j O-n+3 — bn
i bn+3 = O-n
— bn+2 + O-n+1 + bn
+2 + bn+i + an
{
O-n+3 — bbn+3 = an n J O-n+3 ~ bn-ii bn+3 = a„_|
— bn+2 + bn+i + an
n+2 + O-n+1 + bn
— bn+2 + bn+i + bn
T-n+2 + On+1 + 0-n
The first scheme is trivial. All of the others are nontrivial; they have the following recurrent formulas for n > 0:
for T2 : xn+6 = 2.xn+5 + xn+i — 2.xn+3 — xn+2 + xn, for T3 : xn+6 = 2.xn+s — xn+i + 2.xn + 3 - xn+2 - xn, for T4 : xn+e = 2.xn+5 - xn+i + xn+2 + xn+i + xn, for T5 : xn+6 = 3.arn+4 + 2.xn+3 - xn+2 - 2.a:n+i - xn, for T6 : xn+6 — 3.xn+4 + xn+2 + xn,
for T7 : xn+6 = xn+i + A.xn + 3 + xn+2 - xn,
for Tg : xn+6 = £n+4 + 2.xn+3 + 3.xn+2 + 2.a;n+i + a;n. The proofs for these results can be shown by induction, using methods similar to those in Chapter 2.
An open problem is the construction of an explicit formula for each of the schemes given above.