© N. V. Kosinov

Contact: kosinov@unitron.com.ua

The belonging of Fibonacci and Luke’s sequences to the generalized classes of numerical sequences and connection numerical invariants of the generalized sequences with a gold proportion is shown. The Fibonacci and Luke’s numbers are examples of harmonic sequences. The Fibonacci and Luke’s constant of sequences or their numerical invariant is the gold proportion (Ф=1,618033). These sequences are special cases of the generalized algebraic sequence having recurrent property of this kind: a (n) =ka (n-1) +a (n-2) at k =1. If k > 1, constants of sequences are the numbers having properties close to properties of the gold proportion. The equations X±1/X= k allow to receive for sequences the large family numerical invariants, close on properties to the gold proportion having the recurrent ratio a (n) = ka (n-1) ± a (n-2), . The example of a sequence having fractal triad structure is given.

The Fibonacci and Luke’s numerical sequences are examples of harmonic sequences. A constant of Fibonacci and Luke’s sequences or their numerical invariant is the gold proportion (Ф=1,618033). The interest to the gold proportion and Fibonacci and Luke’s numbers is caused not only that the whole areas in culture and science are connected to them, but in the greater degree that they can be found in many phenomena of the environmental world very frequently . Therefore it is important to find out to what generalized classes of Fibonacci and Luke’s sequences the numbers belong, and how the numerical invariants of other sequences are connected to the gold proportion.

Let's consider sequences of this kind:

When k=1 we shall receive the recurrent formula:

This recurrent formula when a (0) = 1, a (1) =1 derives the Fibonacci ’s numbers : Ф (n) = 1, 1, 2, 3, 5, 8, 13 … At a (0) =2 and at a (1) =1 formulas derive Luke’s numbers: L (n) = 2, 1, 3, 4, 7, 11 …

For these sequences the ratio of the next members goes in a limit to the gold proportion:

The number Ф is a numerical invariant or constant of these sequences. The number Ф is the only positive number, which transfers into its opposite when subtracting the unit. The inverse value of the number Ф is exactly equal to the number after a point. The constant of this family of sequences is a root of the equation of this kind: X₁-1/X₁=1.

When k=2 we shall receive the recurrent formula:

This recurrent formula when a (0) = 2 and a (1) =2 derives the main sequence of this kind: K₂ (n) = 2, 2, 6, 14, 34, 82, 198 … In the Neil Sloane’s Encyclopedia this sequence has a registered number A002203 [1].

For sequences of this class:

The number X₂ is a numerical invariant or constant of this family of sequences. The number X₂ has especial property: it transfers into the opposite one when subtracting of the whole part. The inverse value of the number X₂ is precisely equal to the number after a point. In this way the number X₂ shows similarity to the gold proportion. The constant of this family of sequences is a root of the equation of this kind: X₂-1/X₂=2. Any sequence with recurrent property a (n) = 2a (n-1) + a (n-2) with any initial members the ratio of the neighboring members in process of distance from beginning goes to X₂=2,41421356 …

When k=3 we shall receive the recurrent formula:

This recurrent formula when a (0) = 2 and a (1) =3 derives the main sequence of this kind: K₃ (n) = 2, 3, 11, 36, 119, 393, 1298 … In the Neil Sloane’s Encyclopedia this sequence has a registered numberA006497 [1].

For sequences of this class:

The number X₃ is a numerical invariant or constant of these sequences. The number X₃ transfers into its opposite when subtracting the whole part. The inverse value of the number X₃ is precisely equal to the number after a point. The constant of this family of sequences is a root of the equation of this kind: X₃-1/X₃=3.

When k=8 we shall receive the recurrent formula:

This recurrent formula when a (0) = 2 and a (1) =8 derives the main sequence of this kind: K₈ (n) = 2, 8, 66 … In the Neil Sloane Encyclopedia this sequence has a registered number A086594 [1].

For sequences of this class:

The number X₈ is a numerical invariant or constant of these sequences. The number X₈ transfers into its opposite when subtracting the whole part. The inverse value of the number X₈ is precisely equal to the number after a point. The constant of this family of sequences is a root of the equation of this kind: X₈-1/X₈=8.

The equations X-1/X = k allow to receive for sequences the large family of numerical invariants, having the recurrent ratio a (n) = ka (n-1) + a (n-2) similar to the properties of the gold proportion . The examples of numbers with properties similar to properties of the gold proportion are: X₂ = 2.414213562 …, X₃ = 3.302775637 …, X₄ = 4.23606 …, X₅ = 5.192582403 …, X₁₁ = 11.0901 … Their inverse values are given numbers after a point: 1/X₂ = 0.414213562 …, 1/X₃ = 0.302775637 …, 1/X₄ = 0.23606 …, 1/X₅ = 0.192582403 …, 1/X₁₁ = 0.,0901 …

For any k from the formula a (n) =ka (n-1) +a (n-2) when a (0) = 2 and a (1) = k we shall receive an algebraic sequence of this kind: 2, k, k^2+2, k^3+3k, k^4+4k^2+2, k^5+5k^3+5k, k^6+6k^4+9k^2+2, k^7+7k^5+14k^3 +7k, k^8+8k^6+20k^4+16k^2+2, k^9+9k^7+27k^5 +30k^3+9k, k^10+10k^8+35k^6+50k^4+25k^2+2 …

For this algebraic sequence:

The numbers X_k are numerical invariants or constants of the algebraic sequence. The even and odd members of the algebraic sequence are calculated under the Binet’s formulas :

Any members of the algebraic sequence are calculated according the following formula:

The formula: a (n) = ((k+sqrt (k^2+4)) /2) ^n + ((k-sqrt (k^2+4)) /2) ^n gives a large class of the main sequences, first members of which are equal 2, k. Really, zero member of the generalized sequence: a (0) = ((k+sqrt (k^2+4)) /2) ^0 + ((k-sqrt (k^2+4)) /2) ^0 = 2.

The first member of the generalized sequence: a (1) = ((k+sqrt (k^2+4)) /2) ^1 + ((k-sqrt (k^2+4)) /2) ^1 = 2k/2=k, etc.

For these sequences the following formulas are fair:

One algebraic sequence derives a plenty of individual numerical sequences and enables easily to find new sequences of the given class. Having the specific value k the generalized algebraic sequence allows to receive one of specific sequences.

Let's consider sequences of this kind:

When k=1, we shall receive a recurrent formula:

For calculation of first members of the main sequence of this family we shall consider the equation: X₁+1/X₁=1. Let's note, that this equation has no valid roots. The first root of the equation: X₁,1 = (1+sqrt (-3)) /2. The second root of the equation:X_1,2 = (1-sqrt (-3)) /2. The zero member of the sequence: a (0) = ((1+sqrt (-3)) /2) 0 + ((1-sqrt (-3)) /2) 0 =2. The first member of the sequence: a (1) = (1+sqrt (-3)) /2 + (1-sqrt (-3)) /2=1. The second member of the sequence: a (2) = ((1+sqrt (-3)) /2) 2 + ((1-sqrt (-3)) /2) 2 = -1.

The recurrent formula a (n) =a (n-1) - a (n-2) at a (0) = 2 and at a (1) =1 derives numbers:

It is a special numerical sequence. Its feature is that it has triad structure. For this sequence the numerical invariants are members of this sequence. This sequence has unique property - it is fractal. The recurrent formula a (n) = a (n-1) - a (n-2), inducing the fractal sequence, is a symmetric mathematical design in relation to the recurrent formula a (n) = a (n-1) + a (n-2), inducing Luke’s numbers. The equation for calculation of the numerical invariant of the fractal sequence is also symmetrically similar to the equation for the Luke’s sequence.

When k=2, we shall receive the recurrent formula:

This recurrent formula when a (0) = 2 and a (1) =2 derives the main sequence of this kind:

S2 (n) = 2, 2, 2, 2, 2, 2, 2, … In the Neil Sloane’s Encyclopedia this sequence has a registered number A007395 [1].

For sequences of this class:

The number X₂ is numerical invariant or constant of these sequences. The number X₂ has especial property, it transfers into the nearest integer when adding an inverse number to it. The constant of this family of sequences is a root of the equation of this kind: X₂+1/X₂=2.

When k=3, we shall receive the recurrent formula:

This recurrent formula when a (0) = 2 and a (1) =3 derives the main sequence of this kind: K_3- (n) = 2, 3, 7, 18, 47, 123, 322 … In the Neil Sloane’s Encyclopedia this sequence has a registered number A005248 [1].

For sequences of this class:

The number X₃ is a numerical invariant or constant of this family of sequences. The number X₃ transfers into the nearest integer when adding an inverse number. The constant of this family of sequences is a root of the equation of this kind: X₃+1/X₃=3.

When k=8, we shall receive the recurrent formula:

This recurrent formula when a (0) = 2 and a (1) =8 derives the main sequence of this kind: K₈- (n) = 2, 8, 62, 488 … In the Neil Sloane’s Encyclopedia this sequence has a registered number A086903 [1].

For sequences of this class:

The number X₈ is a numerical invariant or constant of this family of sequences. The number X₈ transfers into the nearest integer when adding an inverse number. The constant of this family of sequences is a root of the equation of this kind: X₈+1/X₈=8.

The equations X+1/X=k allow to receive for sequences having the recurrent ratio a (n) = ka (n-1) - a (n-2), the large family of numerical invariants, with the mentioned above properties. Examples of numbers with such properties are: X₂ = 1,00000 …, X₃ = 2,61803 …, X₄ = 3,73205 …, X₅ = 4,79128 …, X₁₁ = 10,9083 … Their inverse values are additions of these numbers up to the nearest integer: 1/ X₂ = 1,000000 …, 1/ X₃ = 0,3819 …, 1/ X₄ = 0,26794 …, 1/X₅ = 0,20871 …, 1/ X₁₁ = 0,0916 …

For any k from the formula a (n) =ka (n-1) - a (n-2) when a (0) = 2 and a (1) = k we shall receive the generalized algebraic sequence of this kind: 2, k, k^2-2, k^3-3k, k^4-4k^2+2, k^5-5k^3+5k, k^6-6k^4+9k^2-2, k^7-7k^5+14k^3-7k, k^8-8k^6+20k^4-16k^2+2, k^9-9k^7+27k^5-30k^3+9k, k^10-10k^8+35k^6-50k^4+25k^2-2 …

For this algebraic sequence:

The numbers X_k- are numerical invariants or constants of the generalized algebraic sequence. The members of an algebraic sequence are calculated using the following formulas:

The formula: a (n) = ((k+sqrt (k^2-4)) /2) ^n + ((k-sqrt (k^2-4)) /2) ^n gives a class of the main sequences. The zero member of an algebraic sequence: a (0) = ((k+sqrt (k^2-4)) /2) ^0 + ((k-sqrt (k^2-4)) /2) ^0 = 2. The first member of an algebraic sequence: a (1) = ((k+sqrt (k^2-4)) /2) ^1 + ((k-sqrt (k^2-4)) /2) ^1 = 2k/2=k. The second member of an algebraic sequence: a (2) = ((k+sqrt (k^2-4)) /2) ^2 + ((k-sqrt (k^2-4)) /2) ^2 = k^2-2 etc.

For these sequences the following formula is fair:

One generalized algebraic sequence derives a plenty of individual sequences and enables easily to find new sequences of the given class. In the table 1 there are given numerical invariants of harmonic of sequences and equation inducing numerical invariants.

Thus, the gold proportion (Ф=1,618033), having a role of the numerical invariant of Fibonacci and Luke’s sequences, is the individual calculation of the generalized equation X-1/X=k when k =1. Accordingly, the Fibonacci and Luke’s sequences of numbers are special cases of the generalized algebraic sequences determined with the generalized recurrent ratio a(n)=ka(n-1) ± a(n-2) when k=1. For the generalized sequences numerical invariants are constants which have similar properties to the properties of the gold proportion. Numerical invariants of harmonic sequences are grouped in families, members of which are submitted by sedate functions of main invariants. Among constants of harmonic sequences there is a whole family of numerical invariants, which are represented with sedate functions of the gold proportion. The recurrent formula a (n) =a (n-1) -a (n-2) when a (0) =2, a (1) =1 symmetric to the formula a (n) =a (n-1) +a (n-2), inducing Luke’s numbers, gives a sequence of this kind: S₁ (n) = 2, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, …, which, to all appearance, has the fundamental status and should be shown in natural phenomena.

1. Neil J.A.Sloane http://www.research.att.com/~njas/sequences/

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