SMARANDACHE PASCAL DERIVED SEQUENCES

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

Given a sequence say S_b . We call it the base sequence. We define a Smarandache Pascal derived sequence S_d as follows:

n

T_n+1 = S ^rC_k .t_k+1 , where t_k is the k^th term of the base sequence.

k=0

Let the terms of the the base sequence S_b be

b_1, b₂ , b₃ , b₄ , . . .

Then the Smarandache Pascal derived Sequence S_d

d_1, d₂ , d₃ , d₄ , . . . is defined as follows:

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

These derived sequences exhibit interesting properties for some base sequences.

Examples:

{1} S_b ® 1, 2, 3, 4, . . . ( natural numbers)

S_d ® 1, 3, 8, 20, 48, 112, 256, . . . ( Smarandache Pascal derived natural number sequence)

The same can be rewritten as

2x2^-1 , 3x2⁰, 4x2¹, 5x2², 6x2³, . . .

It can be verified and then proved easily that T_n = 4( T_n-1 - T_n-2 ) for n > 2.

And also that T_n = (n+1) .2^n-2

{2} S_b ® 1, 3, 5, 7, . . . ( odd numbers)

S_d ® 1, 4, 12, 32, 80, . . .

The first difference 1, 3, 8, 20, 48 , . . . is the same as the S_d for natural numbers.

The sequence S_d can be rewritten as

1.2⁰ , 2.2¹, 3.2², 4.2³, 5.2⁴, . . .

Again we have T_n = 4( T_n-1 - T_n-2 ) for n > 2.Also T_n = n.2^n-1.

{3} Smarandache Pascal Derived Bell Sequence:

Consider the Smarandache Factor Partitions (SFP) sequence for the square free numbers:

( The same as the Bell number sequence.)

S_b ® 1, 1, 2, 5, 15, 52, 203, 877, 4140, . . .

We get the derived sequence as follows

S_d ® 1, 2, 5, 15, 52, 203, 877, 4140 , . . .

The Smarandache Pascal Derived Bell Sequence comes out to be the same. We call it Pascal Self Derived Sequence. This has been established in ref. [1]

In what follows, we shall see that this Transformation applied to Fibonacci Numbers gives beautiful results.

**{4} Smarandache Pascal derived Fibonacci Sequence:

Consider the Fibonacci Sequence as the Base Sequence:

S_b ® 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 114, 233, . . .

We get the following derived sequence

S_d ® 1, 2, 5, 13, 34, 89, 233, . . . ---------- (A)

It can be noticed that the above sequence is made of the alternate (even numbered terms of the sequence ) Fibonacci numbers.

This gives us the following result on the Fibonacci numbers.

n

F_2n = S ⁿC_k .F_k , where F_k is the k^th term of the base Fibonacci sequence.

k=0

Some more interesting properties are given below.

If we take (A) as the base sequence we get the following derived sequence S_dd

S_dd ® 1, 3, 10, 35, 125, 450, 1625, 5875, 21250, . . .

An interesting observation is ,the first two terms are divisible by 5⁰, the next two terms by 5¹ , the next two by 5² , the next two by 5³ and so on.

T_2n º T_2n-1 º 0 ( mod 5ⁿ )

On carrying out this division we get the following sequence i.e.

1, 3, 2, 7, 5, 18, 13, 47, 34, 123, 89, . . . ----------- (B)

The sequence formed by the odd numbered terms is

1, 2, 5, 13, 34, 89, . . .

which is again nothing but S_d ( the base sequence itself.).

Another interesting observation is every even numbered term of (B) is the sum of the two adjacent odd numbered terms. ( 3 = 1+2, 7 = 2 +5, 18 = 5 + 13 etc.)

CONJECTURE: Thus we have the possibility of another beautiful result on the Fibonacci numbers which of-course is yet to be established.

2m+1 r

F_2m+1 = (1/5^m) S { ^2m+1C_r ( S ^rC_k F_k ) }

r =0 k=0

Note: It can be verified that all the above properties hold good for the Lucas sequence ( 1 , 3, 4, 7, 11, . . .) as well.

Pascalisation of Fibonacci sequence with index in arithmetic progression:

Consider the following sequence formed by the Fibonacci numbers whose indexes are in  A. P.

F₁ , F_d+1 , F_2d+1 , F_3d+1 , . . . on pascalisation gives the following sequence

1, d.F₂ , d².F₄ , d³.F₆ , d⁴.F₈ , . . ., dⁿ.F_2n , . . .

for d = 2 and d = 3.

For d = 5 we get the following

Base sequence : F₁, F₆ , F₁₁, F₁₆, . . .

1, 13, 233, 4181, 46368, . . .

Derived sequence: 1, 14, 260, 4920, 93200, . . .in which we notice that

260= 20.(14- 1), 4920 = 20.(260 - 14) , 93200 = 4920 - 260 ) etc . which suggests the possibility of

Conjecture: The terms of the pascal derived sequence for d = 5 are given by

T_n = 20.( T_n-1 - T_n-2 ) ( n > 2)

For d = 8 we get

Base sequence : F_1, F₉, F₁₇, F₂₅ , . . .

S_b ---- 1, 34, 1597, 75025, . . .

S_d ---- 1, 35, 1666, 79919, . . .

= 1, 35, (35-1). 7² , ( 1666 - 35 ). 7² , . . . etc. which suggests the possibility of

Conjecture: The terms of the pascal derived sequence for d = 8 are given by

T_n = 49.( T_n-1 - T_n-2 ) , (n > 2)

Similarly we have Conjectures:

For d = 10 , T_n = 90.( T_n-1 - T_n-2 ) , (n > 2)

For d = 12 , T_n = 18².( T_n-1 - T_n-2 ) , (n > 2)

Note: There seems to be a direct relation between d and the coefficient of ( T_n-1 - T_n-2 ) (or the common factor) of each term which is to be explored.

{5} Smarandache Pascal derived square sequence:

S_b ® 1, 4, 9, 16, 25, . . .

S_d ® 1, 5, 18, 56, 160, 432, . . .

Or 1, 5x1, 6x 3, 7x 8, 8x20, 9x48 , . . . , ( T_n = (n+3)t_n-1 ) , where t_r is the r^th term of Pascal derived natural number sequence.

Also one can derive T_n = 2^n-2 . ( n+3)(n)/ 2 .

{6} Smarandache Pascal derived cube sequence:

S_b ® 1, 8, 27, 64, 125

S_d ® 1, 9, 44, 170, 576, 1792, . . .

We have T_n º 0 ( mod (n+1)).

Similarly we have derived sequences for higher powers which can be analyzed for patterns.

{7} Smarandache Pascal derived Triangular number sequence:

S_b ® 1, 3, 6, 10, 15, 21, . . .

S_d ® 1, 4, 13, 38, 104, 272, . . .

{8} Smarandache Pascal derived Factorial sequence:

S_b ® 1, 2, 6, 24, 120, 720, 5040, . . .

S_d ® 1, 3, 11, 49, 261, 1631, . . .

We can verify that T_n = n.T_n-1 + S T_n-2 + 1.

Problem: Are there infinitely many primes in the above sequence?

Smarandache Pascal derived sequence of the k^th order.

Consider the natural number sequence again:

S_b ® 1, 2, 3, 4, 5, . . . The corresponding derived sequence is

S_d ® 2x2^-1 , 3x2⁰, 4x2¹, 5x2², 6x2³, . . . With this as the base sequence we get the derived sequence denoted by S_d2 as

S_dd or S_d2 ® 1, 4, 15, 54, 189, 648, . . . which can be rewritten as

1, 4x3⁰ , 5x3¹, 6x3² , 7x3³ . . .

similarly we get S_d3 as 1, 5x4^0, 6x4¹, 7x4² , 8x4³ , . . . which suggests the possibility of the terms of S_dk , the k^th order Smarandache Pascal derived natural number sequence being given by

 1, ( k+2) .(k+1)⁰, (k+3).(k+1)¹ , (k+4).(k+1)², . . ., (k+r).(k+1)^r-2 etc. This can be proved by induction.

 We can take an arithmetic progression with the first term 'a' and the common difference 'b' as the base sequence and get the derived k^th order sequences to generalize the above results.

 Reference:[1] Amarnath Murthy, ' Generalization of Partition Function,. Introducing Smarandache Factor Partitions' SNJ, Vol. 11, No. 1-2-3,2000.