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

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. 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-1C_h-1 , ^hC_h-1 , ^h+1C_h-1 , ^h+2C_h-1 , ^h+3C_h-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-1C_h-1 , ^hC_h-1 , ^h+1C_h-1 , ^h+2C_h-1 , ^h+3C_h-1 , . . . where h = 3.2ⁿ.

This can be generalised to conjecture the following:

Conjecture-III :

Given the base sequence as ⁿC_n , ⁿ⁺¹C_n , ⁿ⁺²C_n , ⁿ⁺³C_n , ⁿ⁺⁴C_n , . . .

The SRACS(n) is given by

^h-1C_h-1 , ^hC_h-1 , ^h+1C_h-1 , ^h+2C_h-1 , ^h+3C_h-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

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 T₁ = 2, and T₂ = 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:

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

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.