SMARANDACHE REVERSE AUTO CORRELATED SEQUENCES AND SOME FIBONACCI DERIVED SMARANDACHE SEQUENCES

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

Let a1 , a2 , a3 , . . . be a base sequence. We define a Smarandache Reverse Auto-correlated Sequence (SRACS) b1 , b2 , b3 , . . . as follow :

b1 = a21 , b2 = 2a1a2 , b3 = a22 + 2a1a3 , etc. by the following transformation

n

bn = S ak. an-k+1

k=1

and such a transformation as Smarandache Reverse Auto Correlation Transformation (SRACT)

We consider a few base sequences.

(1) 1 , 2 , 3 , 4 , 5 , . . .

i.e. 1C1 , 2C1 , 3C1 , 4C1 , 5C1 , . . .

The SRACS comes out to be

1 , 4 , 10 , 20 , 35 , . . . which can be rewritten as

i.e. 3C3 , 4C3 , 5C3 , 6C3 , 7C3 , . . . we can call it SRACS(1)

Taking this as the base sequence we get SRACS(2) as

1 , 8 , 36 , 120 , 330, . . . which can be rewritten as

i.e. 7C7 , 8C7 , 9C7 , 10C7 , 11C7 , . . . ,Taking this as the base sequence we get SRACS(3) as

1 , 16 , 136 , 816 , 3876, . . .

i.e. 15C15 , 16C15 , 17C15 , 18C15 , 19C15 , . . . ,

This suggests the possibility of the following :

conjecture-I

The sequence obtained by 'n' times Smarandache Reverse Auto Correlation Transformation (SRACT) of the set of natural numbers is given by the following:

SRACS(n)

h-1Ch-1 , hCh-1 , h+1Ch-1 , h+2Ch-1 , h+3Ch-1 , . . . where h = 2n+1.

2.   Triangular number as the base sequence:

1 , 3 , 6 , 10 , 15 , . . .

i.e. 2C2 , 3C2 , 4C2 , 5C2 , 6C2 , . . .

The SRACS comes out to be

1 , 6 , 21 , 56 , 126 , . . . which can be rewritten as

i.e. 5C5 , 6C5 , 7C5 , 8C5 , 9C5 , . . . we can call it SRACS(1)

Taking this as the base sequence we get SRACS(2) as

1 , 12 , 78 , 364 , 1365, . . .

i.e. 11C11 , 12C11 , 13C11 , 14C11 , 15C11 , . . . ,Taking this as the base sequence we get SRACS(3) as

1 , 24 , 300 , 2600 , 17550, . . .

i.e. 23C23 , 24C23 , 25C23 , 26C23 , 27C23 , . . . ,

This suggests the possibility of the following

conjecture-II

The sequence obtained by 'n' times Smarandache Reverse Auto Correlation transformation (SRACT) of the set of Triangular numbers is given by

SRACS(n)

h-1Ch-1 , hCh-1 , h+1Ch-1 , h+2Ch-1 , h+3Ch-1 , . . . where h = 3.2n.

This can be generalised to conjecture the following:

Conjecture-III :

Given the base sequence as nCn , n+1Cn , n+2Cn , n+3Cn , n+4Cn , . . .

The SRACS(n) is given by

h-1Ch-1 , hCh-1 , h+1Ch-1 , h+2Ch-1 , h+3Ch-1 , . . . where h = (n+1).2n.

SOME FIBONACCI DERIVED SMARANDACHE SEQUENCES

1. Smarandache Fibonacci Binary Sequence (SFBS ):

In Fibonacci Rabbit problem we start with an immature pair ' I ' which matures after one season to 'M' . This mature pair after one season stays alive and breeds a new immature pair and we get the following sequence

I® M ® MI® M IM® M IMMI® MIMMIMIM® MIMMIMIMMIMMI

If we replace I by 0 and M by 1 we get the following binary sequence

0® 1® 10® 101® 10110® 10110101® 1011010110110

The decimal equivalent of the above sequences is

0® 1® 2® 5® 22® 181® 5814

we define the above sequence as the SFBS

We derive a reduction formula for the general term:

From the binary pattern we observe that

Tn = Tn-1 Tn-2 {the digits of the Tn-2 placed to the left of the digits of Tn-1.}

Also the number of digits in Tr is nothing but the rth Fibonacci number by definition . Hence we have

Tn = Tn-1 . 2F(n-2) + Tn-2

Problem: 1. How many of the above sequence are primes?

2. How many of them are Fibonacci numbers?

(2)Smarandache Fibonacci product Sequence:

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, . . .

Take T1 = 2, and T2 = 3 and then Tn = Tn-1 . Tn-2 we get the following sequence

2, 3, 6, 18, 108, 1944, 209952 -------(A)

In the above sequence which is just obtained by the first two terms , the whole Fibonacci sequence  is inherent. This will be clear if we rewrite the above sequence as below:

21, 31, 21 31 , 21 32, 22 33 , 23 35 , 25 38 , . . .

we have Tn = 2Fn-1 . 3Fn

The above idea can be extended by choosing r terms instead of two only and define

Tn = Tn-1 Tn-2 Tn-3. . . Tn-r for n > r.

Conjecture : (1) The following sequence obtained by incrementing the sequence (A) by 1

3, 4, 7, 19, 1945, 209953 . . . contains infinitely many primes .

(2) It does not contain any Fibonacci number.