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, b₂ , b₃ , b₄ , . . .

Then the Smarandache Pascal derived Sequence S_d

d_1, d₂ , d₃ , d₄ , . . . is defined as follows: Ref [1]

d₁ = b₁

d₂ = b₁ + b₂

d₃ = b₁ + 2b₂ + b₃

d₄ = b₁ + 3b₂ + 3b₃ + b₄

. . .

n

d_n+1 = S ⁿC_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

b₁ = d₁

b₂ = -d₁ + d₂

b₃ = d₁ - 2d₂ + d₃

b₄ = -d₁ + 3d₂ - 3d₃ + d₄

. . .

which suggests the possibility of

n

b_n+1 = S (-1)^n+k. ⁿC_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 T₁ = 1 , T₂ = 1 , and T_n-2 = T_n - T_n-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 { T_r is the r^th 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 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 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 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_r is the r^th term of the Fibonacci Sequence.

 (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 , 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