**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 a_{1 }, a_{2} , a_{3}
, . . . be a base sequence. We define a **Smarandache Reverse Auto-correlated
Sequence (SRACS) **b_{1 }, b_{2} , b_{3} , . . . as
follow :

b_{1} = a^{2}_{1} , b_{2}
= 2a_{1}a_{2} , b_{3} = a^{2}_{2} + 2a_{1}a_{3}
, etc. by the following transformation

n

b_{n} = S a_{k}. a_{n-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. ^{1}C_{1 }, ^{2}C_{1
}, ^{3}C_{1 }, ^{4}C_{1 }, ^{5}C_{1
}, . . .

The SRACS comes out to be

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

i.e. ^{3}C_{3 }, ^{4}C_{3
}, ^{5}C_{3 }, ^{6}C_{3 }, ^{7}C_{3
}, . . . 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. ^{7}C_{7 }, ^{8}C_{7
}, ^{9}C_{7 }, ^{10}C_{7 }, ^{11}C_{7
}, . . . ,Taking this as the base sequence we get SRACS(3) as

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

i.e. ^{15}C_{15 }, ^{16}C_{15
}, ^{17}C_{15 }, ^{18}C_{15 }, ^{19}C_{15
}, . . . ,

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-1}**C _{h-1 }, ^{h}C_{h-1 }, ^{h+1}C_{h-1
}, ^{h+2}C_{h-1 }, ^{h+3}C_{h-1 }, . . .
where h = 2^{n+1}.**

**2. ****Triangular number as the base sequence:**

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

i.e. ^{2}C_{2 }, ^{3}C_{2
}, ^{4}C_{2 }, ^{5}C_{2 }, ^{6}C_{2
}, . . .

The SRACS comes out to be

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

i.e. ^{5}C_{5 }, ^{6}C_{5
}, ^{7}C_{5 }, ^{8}C_{5 }, ^{9}C_{5
}, . . . we can call it SRACS(1)

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

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

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

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

i.e. ^{23}C_{23 }, ^{24}C_{23
}, ^{25}C_{23 }, ^{26}C_{23 }, ^{27}C_{23
}, . . . ,

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-1}**C _{h-1 }, ^{h}C_{h-1 }, ^{h+1}C_{h-1
}, ^{h+2}C_{h-1 }, ^{h+3}C_{h-1 }, . . .
where h = 3.2^{n}.**

This can be generalised to conjecture the
following:

**Conjecture-III : **

Given the base sequence as ^{n}C_{n
}, ^{n+1}C_{n }, ^{n+2}C_{n }, ^{n+3}C_{n
}, ^{n+4}C_{n} , . . .

**The SRACS(n) is given by **

^{h-1}**C _{h-1 }, ^{h}C_{h-1 }, ^{h+1}C_{h-1
}, ^{h+2}C_{h-1 }, ^{h+3}C_{h-1 }, . . .
where h = (n+1).2^{n}.**

** **

**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

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

Also the number of digits in T_{r} is
nothing but the r^{th }Fibonacci number by definition . Hence we have

**T _{n} = T_{n-1 . }2^{F(n-2)}
+ T_{n-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 T_{1} = 2, and T_{2} = 3
and then T_{n }= T_{n-1} . T_{n-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:

2^{1}, 3^{1}, 2^{1}
3^{1} , 2^{1} 3^{2}, 2^{2} 3^{3} , 2^{3}
3^{5} , 2^{5} 3^{8} , . . .

we have **T _{n} = 2^{Fn-1} .
3^{Fn}**

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

T_{n }= T_{n-1} T_{n-2}
T_{n-3}. . . T_{n-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.**

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