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 Sb ( called the base sequence).

b1, b2 , b3 , b4 , . . .

Then the Smarandache Pascal derived Sequence Sd

d1, d2 , d3 , d4 , . . . is defined as follows: Ref [1]

d1 = b1

d2 = b1 + b2

d3 = b1 + 2b2 + b3

d4 = b1 + 3b2 + 3b3 + b4

. . .

n

dn+1 = S nCk .bk+1

k=0

Now Given Sd the task ahead is to find out the base sequence Sb . 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

b1 = d1

b2 = -d1 + d2

b3 = d1 - 2d2 + d3

b4 = -d1 + 3d2 - 3d3 + d4

. . .

which suggests the possibility of

n

bn+1 = S (-1)n+k. nCk .dk+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 T1 = 1 , T2 = 1 , and Tn-2 = Tn - Tn-1 , n > 2 -------(A)

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

T0 = T2 - T1 = 0 , T-1 = T1 - T0 = 1 , T-2 = T0 - T-1 = -1 , etc. and we get the Fibonacci sequence extended backwards as follows { Tr is the rth term }

. . . T-6 T-5, T-4, T-3, T-2, T-1, T0, T1, T2, T3, T4, T5, T6, T7, T8, T9, . . .

. . . -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 Sd(-1) comes out to be

Sd(-1) ---- 1, 0, 1, -1, 2, -3, 5, -8, . . .

It can be noticed that , The resulting sequence is nothing but the BEFS rotated by 1800 about T1 . and then the terms to the left of T1 omitted. { This has been generalised in the Proposition 2 below.}

It is not over here. If we further depascalise the above sequence we get the following sequence Sd(-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 F2r+2 , F2r+4 , F2r+6 , F2r+8 , . . . where Fr is the rth term of the Fibonacci Sequence.

(2) In the FEBS If we take Tr 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 Tr as the first term.

As an example let r = 7 , T7 = 13

. . . T-6 T-5, T-4, T-3, T-2 , T-1 , T0, T1, T2, T3, T4, T5, T6, T7, T8, T9, . . .

. . . -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 (T7) by 1800 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