COUPLED RECURRENCE RELATIONS
The 2-Fibonacci Sequences
4.2 Fibonacci sequence via arithmetic progression
The idea for this research was generated by Marchisotto's paper [23]. Thus we borrowed the first part of its title and invited colleagues to prepare a series of papers under the first part of this title.
Here we shall discuss an approach for an interpretation of the Fibonacci sequence as an arithmetic progression. The reasoning for this is the fact that there is a relation between the way of generating the Fibonacci sequence and the way of generating the arithmetic progression. On the other hand, obviously, the Fibonacci sequence is not an ordinary arithmetic progression.
Thus we can construct a new type of progression which will include both the ordinary arithmetic progression, and the Fibonacci sequences (the classical one and its generalizations.
Let / : N —>• R be a fixed function, where N and R are the sets of the natural and real numbers, respectively, and a be a fixed real number. The sequence
a,a + f(l),a + f(2),...,a + f(k),... (4.1) we shall call an A-progression (from 'arithmetic progression').
Obviously, if ak = a + f(k) is its k-th member, then E ak
=
(n+
l).a+ £/(A).
fc=0 k=0
When f(k) = k.d for the fixed real number d we obtain from (4.1) the ordinary arithmetic progression.
When a = 0 and / is the function defined by:
/ ( l ) - 1, /(2) = 1, f(k + 2) = / ( * + 1) + f(k) for A > 1, we obtain from (4.1) the ordinary Fibonacci sequence. Therefore, the ordi- nary Fibonacci sequence can be represented by an A-progression. We shall show that some of the generalizations of this sequence can be represented by an A-progression, too. When a and b are fixed real numbers and / is a function defined by
/ ( l ) = b - a, /(2) = 6, f(k + 2) = / ( * + 1) + f(k) + a,
we obtain from (4.1) the generalized Fibonacci sequence a,b,a + b, a + 2.6, 2.o + 3.6, ....
When a, b and c are fixed real numbers and / is a function defined by / ( l ) = 6 - o , f(2) = c-a, f(3)=b + c,
f(k + 3) = f{k + 2) + f(k + 1) + f(k) + 2.a,
we obtain from (4.1) the generalized Fibonacci sequence sometimes known as the Tribonacci sequence: a, b, c, a + b + c, a + 2.6 + 2.c, 2.a + 3.6 + 4.c,...
When a, 6, c and d are fixed real numbers, and / and g are functions defined by:
/ ( l ) = -a + b, f{2) = -a + c + d, f(k + 2) = g(k + 1) + g(k) - a + 2.c (k > 1)
5(1) = -c + d, g(2) = a + b-c, g(k + 2) = f(k + l) + f{k) + 2.a-c (k > 1)
we obtain from (4.1) the first of the generalizations of the Fibonacci se- quence from Chapter 2. When for the same a, b, c and d
/ ( l ) = -a + b, f{2) = -a + b + c, f(k + 2) = f(k + l)+g(k)+c(k>l)
S(l) = -c + d, g(2)=a-c + d, g(k + 2) = g(k + l) + f(k) + a(k>l)
we obtain from (4.1) the second of the generalizations of the Fibonacci sequence from Chapter 2.
The above idea for combination of elements of different mathematical areas with Fibonacci numbers founded a realization in the following short research, too.
Let {cti}il0 be a sequence with real numbers. We can construct a new sequence {/?j}£^0 related to the first one, which is an analogy (and extension) of the arithmetic progression, following the scheme:
b0 = a0
k (4 2)
h = bk-i + £ ffli-
f=i
For example, if ao = 0, a\ = a-i — ... — 1, we obtain the sequence
b0 — 0,6i = 1,&2 = 3, ...,bk = k.(k + l ) / 2 = ifc(A;-th triangle number) for 1 < k.
n
Let Sn = J2 bk-
fc=i
The following assertion can be proved directly by induction:
Theorem 4.1: For every natural number n:
(a) bn — a0 + £ (n + l-k).ak,
k-l
(4.3) (b) Sn = n.o0 + X) in+i-fc-afc-
fc=i
We can see that dk = bk — bk~i — Y^ ai a n^ dh — ° k - i — ak-
Therefore we obtain a situation which is analogous to acceleration in mechanics (in the sense of a velocity of a velocity). In the particular case, when a2 = as = ... = 0, we obtain the ordinary arithmetic progression.
The extension of the concept 'arithmetic progression' introduced in [3], which has the b-form of the following sequence:
b, b + d, b + 2.d, ...,b + p.d, b + p.d + e,b + p.d + 2.e,...,
b + p.d + q.e, b + (p + l).d + q.e, ...,b + 2.p.d + q.e, b + 2.p.d + (q + l).e,...
also can be represented in the above form by the o-sequence b, d, d, ...,d, e,e, ...,e,d, d, ...,d, e,....
p times q times p times
When a sequence {/?j}£fi0 is given, we can construct the sequence {ai}^l0
from the formulae (see (4.2)):
ao = bo
k (4 4)
ak - h - Z) {n + 1 - k).ai7
i=l
where a, are previously calculated members of { a j } g0 .
Therefore, we can define a function F, which juxtaposes to the sequence {ati}iZo ^ e sequence {/3i}?Z0, or briefly, F(a) = b. If all members of the sequence b are members of sequence a, after a finite member of initial members, then we say that b is a sequence autogenerated by a. It can also
be easily seen that the sequence {oii}^0 for which a,i = 0 (0 < i < oo) is the unique fixed point of F.
As shown above, the ordinary arithmetic progression is not autogener- ated (in the general case). Below we shall construct a sequence which is autogenerate in a special sense.
Let ao = l , a i = 0,a2 = 1,0:3 — 0,0:4 = 1,05 = 1,Q!6 = 2,07 = 3, etc.
(after the first 3 elements, all other members of this sequence are the mem- bers of the Fibonacci sequence). Then the sequence { f t } ^0 has the form 1,1,2,3,..., i.e. the same Fibonacci sequence without its 0-th member. We can construct another a- sequence which generates the Fibonacci sequence as its 6-sequence.
Let ao = 0 , a i = 1,02 == ~ l>c*3 — l,a!4 = 0,0:5 = 1,06 = 1,0:7 = 2, etc. (after the first four elements, all other members of this sequence are the members of the Fibonacci sequence). Therefore, the Fibonacci sequence {Fjj^Q is an autogenerated one. From here it can easily be seen that the following equality (see [20]) is valid for every natural number n:
n
F
n+4= n + 3+ D (n + 1-*)•**•
fc=i
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Part A, Section 1
Coupled Recurrence Relations
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