13 August 2024

St. Pontian 236 AD; Bl. Philip Monarriz & Comps. 1936 AD; St. Hippolytus 235 AD; St. Casian 3^rd century AD; St. Radegund 587 AD

This article will be about some of the properties of the Multiplication/Division Tables and Transcendental Fibonacci Sequence. There is a way of breaking a Transcendental Fibonacci Sequence into an arithmetic formula and in that way obtaining all the terms of this sequence. To do this, I have to bring back certain properties of the Multiplication/Division Table. We started with two numbers:

 

P(ater) = (7/5)^1/2 * π = 3.71718255693

 

And

 

F(ilius) = (5/7)^1/2 * e = 2.29736745287

 

We can obtain the third essential, the parent number, by division of P/F:

 

S(piritus) = P / F = (7/5) * (π / e) = 1.61801828971

 

The classical Fibonacci Golden Ratio is good for rough calculations of the properties of bigger objects, architecture, nature's patterns, etc. However, the Transcendental Fibonacci formula is superior in accuracy.

Classical Fibonacci Golden Ratio:

 

Classical Fibonacci Golden Ratio = (5^1/2 + 1) / 2 = 1.61803398875

 

Now, to calculate more of these transcendental numbers upwards (not proven yet if they are transcendental indeed), we multiply factors of one pair to obtain the result, i.e., third pair; and to calculate transcendentals downwards, we divide two factors obtaining the third entry. In other words:

Transcendentals upwards:

STx = F * P

FTx = P * STx

PTx = STx * FTx

Etc.

Transcendentals downwards:

PVx = F / S

FVx = S / PVx

SVx = PVx / FVx

Etc.

 

Here is the link to the previous article with detailed calculations for ‘Constants going down’

>>>   https://luxdeluce.com/357-145-division-multiplication-table-of-transcendental-constants-in-quantum-physics-and-cosmology.html

And detailed calculations for ‘Constants going up’

>>>   https://luxdeluce.com/356-144-multiplication-division-table-of-transcendental-constants-in-quantum-physics-and-cosmology.html

 

To obtain the Transcendental Fibonacci Sequence using an arithmetical formula, we need one pair just below the entry S, i.e., FVx and PVx.

And the formula for the terms of Transcendental Fibonacci going up is:

 

S_n+1 = [(FVx)⁽ⁿ⁺¹⁾] * [(PVx)ⁿ]       Eqn. 1

 

Where

S_n+1 is the term of the Transcendental Fibonacci Sequence

FVx = 1.13955788072 (from the tables; see link above)

PVx = 1.41986494682 (from the tables; see link above)

n >= 0; n are the integers

 

You will see from the calculations that the ratio (‘golden ratio’) of two consecutive terms is equal to:

 

GR = 1.61801828971

 

Another important observation is that the terms of the Transcendental Fibonacci Sequence are exact values, they can be rounded up to the nearest integer as well, but only the first six terms are the same as in the case of the classical Fibonacci sequence, i.e., 1, 2, 3, 5, 8, 13; the next terms are slightly smaller than in classical Fibonacci sequence.

 

Here are the calculations of the first fifteen terms:

 

S_n+1 = [(FVx)⁽ⁿ⁺¹⁾] * [(PVx)ⁿ]       Eqn. 1

 

Where

S_n+1 is the term of the Transcendental Fibonacci Sequence

FVx = 1.13955788072 (from the tables; see link above)

PVx = 1.41986494682 (from the tables; see link above)

n >= 0; n are the integers

 

n=0;    S₀₊₁ = S₁ = (1.13955788072)⁰⁺¹ * (1.41986494682)⁰ = 1.13955788072 = approx. 1

n=1     S₁₊₁ = S₂ = (1.13955788072)¹⁺¹ * (1.41986494682)¹ = 1.84382549318 = approx. 2

n=2     S₂₊₁ = S₃ = (1.13955788072)²⁺¹ * (1.41986494682)² = 2.983343371 = approx. 3

n=3     S₃₊₁ = S₄ = (1.13955788072)³⁺¹ * (1.41986494682)³ = 4.82710413875 = approx. 5

n=4     S₄₊₁ = S₅ = (1.13955788072)⁴⁺¹ * (1.41986494682)⁴ = 7.81034278281 = approx. 8

n=5     S₅₊₁ = S₆ = (1.13955788072)⁵⁺¹ * (1.41986494682)⁵ = 12.6372774715 = approx. 13

n=6     S₆₊₁ = S₇ = (1.13955788072)⁶⁺¹ * (1.41986494682)⁶ = 20.4473460809

n=7     S₇₊₁ = S₈ = (1.13955788072)⁷⁺¹ * (1.41986494682)⁷ = 33.084179935

n=8     S₈₊₁ = S₉ = (1.13955788072)⁸⁺¹ * (1.41986494682)⁸ = 53.5308082347

n=9     S₉₊₁ = S₁₀ = (1.13955788072)⁹⁺¹ * (1.41986494682)⁹ = 86.6138267866

n=10   S₁₀₊₁ = S₁₁ = (1.13955788072)¹⁰⁺¹ * (1.41986494682)¹⁰ = 140.142755882

n=11   S₁₁₊₁ = S₁₂ = (1.13955788072)¹¹⁺¹ * (1.41986494682)¹¹ = 226.753542187

n=12   S₁₂₊₁ = S₁₃ = (1.13955788072)¹²⁺¹ * (1.41986494682)¹² = 366.891378515

n=13   S₁₃₊₁ = S₁₄ = (1.13955788072)¹³⁺¹ * (1.41986494682)¹³ = 593.636960772

n=14   S₁₄₊₁ = S₁₅ = (1.13955788072)¹⁴⁺¹ * (1.41986494682)¹⁴ = 960.515459976

 

The ratio (‘golden ratio’) of the two consecutive numbers is equal to

 

GR = 1.61801828971

 

Next, we shall discuss the Fractional Fibonacci Sequence and the complete Transcendental Fibonacci Sequence.

 

Miraculous picture of Our Lady of Absam, 17 January 1797 AD, St. Michael’s Basilica, Absam, Tyrol, Austria:

 

 

Schubert: Ave Maria - Elisabeth Kulman   >>>   https://www.youtube.com/watch?v=Rw9DueQot48