**DEPASCALISATION OF SMARANDACHE
PASCAL DERIVED SEQUENCES **

**AND BACKWARD EXTENDED FIBONACCI
SEQUENCE**

Amarnath Murthy, S.E.(E&T) , WLS, Oil and Natural Gas Corporation Ltd., Sabarmati, Ahmedabad,- 380005 INDIA.

Given a sequence **S _{b}** ( called the base sequence).

b_{1, }b_{2 }, b_{3} , b_{4} , . . .

Then the Smarandache Pascal derived Sequence **S _{d}**

d_{1, }d_{2 }, d_{3} , d_{4} , . . . is
defined as follows: **Ref [1]**

d_{1} = b_{1}

d_{2} = b_{1} + b_{2}

d_{3} = b_{1} + 2b_{2} + b_{3}

d_{4} = b_{1} + 3b_{2} + 3b_{3} + b_{4}

**. . .**

n

d_{n+1} = S ^{n}C_{k}
.b_{k+1}

k=0

Now Given S_{d }the task ahead is to find out the base sequence S_{b}
. We call the process of extracting the base sequence from the Pascal derived
sequence as **Depascalisation.** The interesting observation is that this
again involves the Pascal's triangle though with a difference.

We see that

b_{1} = d_{1}

b_{2} = -d_{1} + d_{2}

b_{3} = d_{1} - 2d_{2} + d_{3}

b_{4} = -d_{1} + 3d_{2} - 3d_{3} + d_{4}

**. . .**

which suggests the possibility of

n

b_{n+1} = S (-1)^{n+k}.
^{n}C_{k} .d_{k+1}

k=0

This can be established by induction.

We shall see that the depascalised sequences also exhibit interesting patterns.

To begin with we define The **Backward Extended Fibonacci Sequence** **(BEFS)**
as Follows:

The Fibonacci sequence is

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .

In which T_{1} = 1 , T_{2} = 1 , and T_{n-2}** ^{
}**= T

Now If we allow n to take values 0 , -1 , -2 , . . . also , we get

T_{0} = T_{2} - T_{1} = 0 , T_{-1} = T_{1}
- T_{0} = 1 , T_{-2} = T_{0} - T_{-1} = -1 ,
etc. and we get the Fibonacci sequence extended backwards as follows { T_{r}
is the r^{th }term }

**. . .** T_{-6} T_{-5}, T_{-4}, T_{-3},
T_{-2}, T_{-1}, ** T_{0}**, T

**. . .** -8, 5, -3, 2 , -1, 1 ** 0**, 1, 1, 2 , 3, 5 8, 13, 21,
34,

**1.
****Depascalisation of the Fibonacci sequence:**

The Fibonacci sequence is

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . .

The corresponding depascalised sequence S_{d(-1) }comes out to be

S_{d(-1) }---- 1, 0, 1, -1, 2, -3, 5, -8, . . .

It can be noticed that , The resulting sequence is nothing but the **BEFS
rotated by 180 ^{0} about T_{1} . **and then the terms to the
left of T

It is not over here. If we further depascalise the above sequence we get the
following sequence S_{d(-2)} as

1 , -1, 2, -5, 13, -34 , 89 , -233

This can be obtained alternately from the Fibonacci Sequence by:

a. Removing even numbered terms.

b. Multiplying alternate terms with (-1) in the thus obtained sequence.

**Propositions:**

Following two propositions are conjectured on Pascalisation and Depascalisation of Fibonacci Sequence.

**(1)** If the first r terms of the Fibonacci Sequence are removed and
the remaining sequence is Pascalised , the resulting Derived Sequence is **F _{2r+2
}, F_{2r+4 }, F_{2r+6 }, F_{2r+8 }, . . . **where
F

** (2) **In the **FEBS** If we take T_{r }as the first
term and Depascalise the Right side of it then we get the resulting sequence as
the left side of it ( looking rightwards) with T_{r }as the first term.

As an example let r = 7 , T_{7} = 13

. . . *T _{-6} T_{-5}, T_{-4}, T_{-3}, T_{-2
}, T_{-1 }, T_{0}, T_{1}, T_{2}, T_{3},
T_{4}, T_{5}, T_{6},*

. . *. -8, 5, -3, 2 , -1, 1 0, 1, 1, 2 , 3, 5 8,* ** 13**, 21,
34, 55, 89, . . .

® ® ® ® ® ® ® ® ® ®

depascalisation

The Depascalised sequence is

13, 8, 5, 3, 2, 1, 1, 0, 1, -1, 2, -3, 5, -8 . . .

which is obtained by rotating the FEBS around 13
(T_{7}) by 180^{0} and then removing the terms on the left side
of 13.

One can explore for more fascinating results.

**References:**

**[1] "**Amarnath Murthy" ,
'Smarandache Pascal derived Sequences', SNJ , 2000