155. The Fractional & Complete Transcendental Fibonacci Sequence from Multiplication/Division Tables.

23 July 2024 AD

St. Rose of Lima 1617 AD; St. Philip Benizi 1285 AD

 

Below is the Complete Transcendental Fibonacci Sequence consisting of two parts, whole and fractional. The terms are bounded by zero and infinity. The ratio of two consecutive terms is equal to the ‘golden ratio:’

GR = 1.61801828971…

The Whole Transcendental Fibonacci Sequence from n=0 to n=14, with terms going up to infinity:

The algebraic formula for the Whole Transcendental Fibonacci Sequence:

Sn+1 = (FVx)n+1 * (PVx)n  ; n>=0     Eqn. 1

FVx = 1.13955788072

PVx = 1.41986494682

 

And so on…

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

The Fractional Transcendental Fibonacci Sequence from n= 0 to n=-15 with terms going down to zero:

The algebraic formula for the Fractional Transcendental Fibonacci Sequence:

Sn = (FVx)n+1 * (PVx)n   ; n=<0    Eqn. 2

FVx = 1.13955788072

PVx = 1.41986494682

 

n=0     S0     = (1.13955788072)0+1 * (1.41986494682)0 = 1.13955788072

n=-1    S-1    = (1.13955788072)-1+1 * (1.41986494682)-1 = 0.704292335859

n=-2    S-2    = (1.13955788072)-2+1 * (1.41986494682)-2 = 0.435280824907

n=-3    S-3    = (1.13955788072)-3+1 * (1.41986494682)-3 = 0.269020954631

n=-4    S-4    = (1.13955788072)-4+1 * (1.41986494682)-4 = 0.166265806848

n=-5    S-5    = (1.13955788072)-5+1 * (1.41986494682)-5 = 0.102758855018

n=-6    S-6    = (1.13955788072)-6+1 * (1.41986494682)-6 = 0.0635090812455

n=-7    S-7    = (1.13955788072)-7+1 * (1.41986494682)-7 = 0.03925115164

n=-8    S-8    = (1.13955788072)-8+1 * (1.41986494682)-8 = 0.0242587811829

n=-9    S-9    = (1.13955788072)-9+1 * (1.41986494682)-9 = 0.0149928967659

n=-10   S-10 = (1.13955788072)-10+1 * (1.41986494682)-10 = 0.00926620969694

n=-11   S-11 = (1.13955788072)-11+1 * (1.41986494682)-11 = 0.00572688810496

n=-12   S-12 = (1.13955788072)-12+1 * (1.41986494682)-12 = 0.00353944584026

n=-13   S-13 = (1.13955788072)-13+1 * (1.41986494682)-13 = 0.00218751905512

n=-14   S-14 = (1.13955788072)-14+1 * (1.41986494682)-14 = 0.00135197424469

n=-15   S-15 = (1.13955788072)-15+1 * (1.41986494682)-15 = 0.000835574142324

And so on…

 

Obviously, the two parts of the Transcendental Fibonacci Sequence can be merged together to form one complete sequence.

 

Suggested pages:

https://en.wikipedia.org/wiki/Fibonacci_sequence

https://en.wikipedia.org/wiki/Lucas_number

https://en.wikipedia.org/wiki/Golden_ratio

 

 

Next, I will write two more examples of Parent Transcendental Constants, and then go to the Transcendental Fibonacci Sequences based on addition/subtraction tables, so we will see the difference between the ‘cross product’ (multiplication/division tables) and the ‘dot product’ (addition/subtraction tables).

 

 

 

 

 

 

Paolo Veneziano - Coronation of the Virgin Polyptych, central panel, ca. 1350, tempera & gold, 167 x 285 cm, Gallerie dell'Accademia, Venice, Italy:

 

 

 

 

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