Download Sequence t-Balancing Numbers

Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 47, 2305 - 2310
Sequence t-Balancing Numbers
K. K. Dash and R. S. Ota
Department of Mathematics, Government College
Sundargarh-770 002, Odisha, India
[email protected]
[email protected]
Sarthak Dash
BITS-Pilani – 333031, India
[email protected]
Abstract
The concept of t-balancing numbers is generalized to an arbitrary sequence;
thereby sequence t-balancing numbers are introduced and defined.
Mathematics Subject Classification: 11D25, 11D41
Keywords: Balancing number, Co-balancing number, Sequence Balancing number,
Sequence Co-balancing number, t-Balancing number, Sequence t-Balancing number.
1. INTRODUCTION
A positive integer n is called a co-balancing number [8] if 1 2 ⋯ … … 1
2
… … … for some positive integer r called the
co-balancer and balancing number [1] if 1 2 ⋯ … … 1 1
2
… … … where r is called balancer.
2306
K. K. Dash, R. S. Ota and S. Dash
We define a number n to be a t-balancing number if 1 2 … … … 1 2 … … … , r is called the t-balancer. With
this generalized definition, the 0- balancing numbers are the co-balancing numbers
and 1- balancing numbers are one less than the balancing numbers.
For a sequence of real numbers ∞
, Panda [6] defined a number of
this sequence a sequence balancing number if
… … … … … … for some natural number r. Similarly, he defined a sequence cobalancing
number if
… … … … … … for some natural number r.
We generalize the above concepts as sequence t-balancing number.
We call a number a sequence t-balancing number if
… … … … … … Accordingly, the sequence 0-balancing number is the sequence cobalancing number
and the sequence 1-balancing number is the next term of sequence balancing
number.
2. SEQUENCE t-BALANCING NUMBERS IN CERTAIN SEQUENCES
In this section we investigate sequence t-balancing numbers in some real
sequences. Throughout this section we denote as the nth t-balancing number and
as the nth t-balancer, where , ∈ .
2.1 SEQUENCE t-BALANCING NUMBERS IN THE SEQUENCE OF
ODD NATURAL NUMBERS.
Let 2 1. Then any sequence t-balancing number 2 1 of this
sequence satisfies
1 3 ⋯ 2 1
2 2 1
2 2 3
⋯ 2 2 2 1
For example 16,21,31,33,43 are sequence 0-balancing numbers with sequence 0balancers 7,9,13,14,18 respectively having sum 31,41,61,65,85 in each cases.
3,15,20,24,32 are sequence 1-balancing numbers with sequence 1-balancers
1,6,8,10,13 respectively having sum 5,29,39,47,53 in each cases. 6,8,13,18,23 are
sequence 2-balancing numbers with sequence 2-balancers 2,3,5,7,9 respectively
having sum 11,15,25,35,45 in each cases. 9,11,16,18,21 are sequence 3-balancing
numbers with sequence 3-balancers 3,4,6,7,8 respectively having sum 17,21,31,35,41
Sequence t-balancing numbers
2307
in each cases. 9,12,14,16,18 are sequence 4-balancing numbers with sequence 4balancers 3,4,5,6,7 respectively having sum 17,23,27,31,37 in each cases.
15,17,19,29,39 are sequence 5-balancing numbers with sequence 5-balancers
5,6,7,11,15 respectively having sum 29,33,37,57,77 in each cases.
Theorem 2.1.1:- The recurrence relation for the sequence t-balancing number in the
sequence of odd natural numbers is ! 6!#$ !#% 4 1
.
Proof :- We have 2 1. Then any sequence t-balancing number 2 1 of
this sequence satisfies
1 3 ⋯ 2 1
2 2 1
2 2 3
⋯ 2 2 2 1
⟹ ⟹ √2 2
.…… (1)
Now since 2 1 is a sequence t-balancing number, therefore 2 2
must be a perfect square. Let 2 2 ) , for some y. Modifying this equation,
we get, * 2) , where * 2 and we need to find integer solution to this
equation. Once we get x, we can obtain * and putting r in equation (1) to
get m and hence 2 1, which will be the sequence t-balancing number.
Now consider the equation * 2) ………. (2)
So, we need to find integral solution of the above generalized Pell’s equation (2).
Clearly *, )
3, 2
solves equation (2).
Consider the related Pell’s equation * 2) 1
……….. (3)
The fundamental solution of this equation is *, )
3,2
. Therefore the general
solution* , ) is given by
* + {3 2√2
3 2√2
}
) + √{3 2√2
3 2√2
} ..……. (4)
We need to find the primitive solutions for * 2) . If *, )
are the
primitive solutions for the above equation, then we have
, ** + 2)) -. *) + )* ..…… (5)
are the generic solution for * 2) . Since 3, 2
is one of the primitive
solution, therefore
, 3* + 4) and . 3) + 2*
forms one set of solutions for the required equation. Now substituting the values of
* -) in the above expressions, we have
0
1
0
1
, =/+ + √2 3 2√2
/∓ + √2 3 2√2
and
and
0
0
. = /+ √ + 2 3 2√2
/+ √ ∓ 2 3 2√2
…… (6)
K. K. Dash, R. S. Ota and S. Dash
2308
Therefore, rewriting the above expressions, we get two sets of , , . expressions,
which are as follows
2, +43 2√2
3 2√2
5,
2√2. +43 2√2
3 2√2
5
.....…. (7)
and
2, +43 2√2
3 2√2
5,
2√2. +43 2√2
3 2√2
5, 6 1 ......… (8)
Since, the problem definition tells us that both , -. are positive, therefore the
following two sets of expressions for , -. satisfy the requirements.
2, 783 2√29
2√2. 783 2√29
83 2√29
83 2√29
:,
:
……… (9)
And
2,; 43 2√2
3 2√2
5,
2√2.; 43 2√2
3 2√2
5
…..….. (10)
;
Hence from the above expressions, we find that , , and . .;. Therefore
the combined non-trivial solutions of * 2) is defined as per equation (9).
Also, we have , -. as defined in (9) satisfy , 6, , and . 6. . respectively.
Now, we have that , . i.e. m (index of the sequence t-balancing
number) is expressed as a function of n. So, substituting the recurrence relations of
, -. in this expression, we get a recurrence relation for the index,
6 2
………… (11)
So, now using (11) and the fact that 2 1, we get the following recurrence
relation for the sequence t-balancing numbers within the sequence of odd integers,
! 6!#$ !#% 4 1
.
∎
2.2 SEQUENCE t-BALANCING
SEQUENCE :
NUMBERS
IN
THE
FIBONACCI
A sequence t-balancing number = in the Fibonacci sequence would satisfy:
= = ⋯ = = = ⋯ =
for some t and r. But it is well known that
= > = = ⋯ = > = =
Sequence t-balancing numbers
2309
for ? 2, it follows that no Fibonacci number = for ? 2 can be a sequence tbalancing number. For @ 2, we have
= = 1 1 2 =0 .
Hence, the only sequence 0-balancing number in the Fibonacci sequence is = 1.
The above discussion proves the following theorem.
Theorem 2.2.1:- The only sequence t-balancing number in the Fibonacci sequence
for t = 0 is = 1.
REFERENCES
[1] BEHERA, A., PANDA,G.K., On the square roots of triangular numbers,
Fibonacci Quarterly, 37 No.2(1999) 98-105.
[2] LIPTAI, K., Fibonacci balancing numbers, Fibonacci Quarterly, 42 No.
4(2004)330-310
[3] LIPTAI, K., Lucas balancing numbers, Acta Math.Univ.Ostrav., 14 No. 1 (2006)
43-47
[4] LIPTAI, K., LUCA F., PINTER, A., SZALAY L., Generalized balancing
numbers, Indagationes Math. N.S., 20(2009) 87-100
[5] OLAJOS, P., Properties of balancing, cobalancing and generalized balancing
numbers, Annales Mathematicae et Informaticae, 37 (2010) 125-138
[6] PANDA, G.K., Sequence balancing and cobalancing numbers, Fibonacci
Quarterly, 45 (2007)265-271
[7] PANDA, G.K., Some fascinating properties of balancing numbers, Proceedings of
the Eleventh International Conference on Fibonacci Numbers and their Applications,
Cong.Numer. 194 (2009) 185-189
[8] PANDA,G.K., RAY,P.K.,Cobalancing numbers and cobalancers, Int. J. Math.
Sci.,No.8(2005)1189-1200
2310
K. K. Dash, R. S. Ota and S. Dash
[9] PANDA, G.K., RAY, P.K., Some links of balancing and cobalancing numbers
and with Pell and associated Pell numbers, (oral communication).
Received: September, 2012