156. The Two Parent Transcendental Fibonacci-like Sequences, Whole and Fractional, and Partial Transcendental Number System.

30 August 2024 AD

St. Rose of Lima 1617 AD; Sts. Felix & Adauctus 304 AD; St. Fiacre of Brie 670 AD; Bl. Bronislava 1259 AD

The Transcendental Fibonacci Sequence is the result of operation on two Parent Sequences (or if you prefer Transcendental Constants, more about it in a moment), the P(ater) constant and F(ilius) constant.

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

And

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

The division of P(ater) / F(ilius) is equal to

S(piritus) = [ ( 7 / 5 )1/2 * π ] / [ ( 5 / 7 )1/2 * e ] = ( 7 / 5 ) * ( π / e ) = 1.61801828971

This result is equal to the Transcendental Fibonacci Ratio, and the Sequence can be derived from it.

Connection between a transcendental constant and a sequence (algebraic representation of a constant) and a spiral (geometric representation of a transcendental constant).

Each constant in the multiplication/division tables and the addition/subtraction tables represents the ‘Golden Ratio’, that is, the unique number assigned to each sequence/spiral. The most known will be the Fibonacci Golden Ratio, however, each sequence has its own ‘golden ratio.’ The ‘golden ratio’ irrational number immediately suggests the presence of a sequence, as an algebraic representation of the golden ratio and a spiral, which is a geometric representation of a sequence/’golden ratio.’

So, each entry in the multiplication/division tables can be represented by a sequence (algebraic representation) or a spiral (geometric representation). These entries create the space of never-ending sequences/spirals. Can it be called the Transcendental Space or Number System? Similar to algebraic (man-made) numbers? If so, then they would fill the void of having just two basic transcendental numbers, π and e, plus a couple of the less known ones. These known transcendentals do not form any ordered system. On the contrary, the entries of the multiplication/division tables and the addition/subtraction tables are very well ordered and have a lot of beautiful properties that I will mention later.

Let us move to the F(ilius) and P(ater) sequences among infinity of other ones. Both will have similar formulas, but the ‘golden ratio’ will be different for each of them. I will write some of the terms of a sequence. Just to remind you, each sequence goes from zero to plus infinity for multiplication/division tables.

 

F(ilius) Sequence.

Three consecutive entries are taken into account, PVx, Sx, and F.

Fn+1 = (PVx)n+1 * (S)n    n >= 0       Eqn. 1

And

Fn = (PVx)n+1 * (S)n    n =< 0       Eqn. 2

 

‘golden ratio’, F = 2.29736745287

PVx = 1.41986494682

S = 1.61801828971

 

And so on…

n=12;  F12+1 = F13 = (1.41986494682)12+1 * (1.61801828971)12 = 30691.1107487

n=11;  F11+1 = F12 = (1.41986494682)11+1 * (1.61801828971)11 = 13359.252004

n=10;  F10+1 = F11 = (1.41986494682)10+1 * (1.61801828971)10 = 5815.02623243

n=9;    F9+1 = F10 = (1.41986494682)9+1 * (1.61801828971)9 = 2531.16941531

n=8;    F8+1 = F9 = (1.41986494682)8+1 * (1.61801828971)8 = 1101.76951107

n=7;    F7+1 = F8 = (1.41986494682)7+1 * (1.61801828971)7 = 479.579141637

n=6;    F6+1 = F7 = (1.41986494682)6+1 * (1.61801828971)6 = 208.75160438

n=5;    F5+1 = F6 = (1.41986494682)5+1 * (1.61801828971)5 = 90.8655705549

n=4;    F4+1 = F5 = (1.41986494682)4+1 * (1.61801828971)4 = 39.5520405067

n=3;    F3+1 = F4 = (1.41986494682)3+1 * (1.61801828971)3 = 17.2162448185

n=2;    F2+1 = F3 = (1.41986494682)2+1 * (1.61801828971)2 = 7.4939012464

n=1;    F1+1 = F2 = (1.41986494682)1+1 * (1.61801828971)1 = 3.2619515163

n=0;    F0+1 = F1 = (1.41986494682)0+1 * (1.61801828971)0 = 1.41986494682

 

n=0;    F0 = (1.41986494682)0+1 * (1.61801828971)0  = 1.41986494682

n=-1;    F-1 = (1.41986494682)-1+1 * (1.61801828971)-1 = 0.618039985308

n=-2;    F-2 = (1.41986494682)-2+1 * (1.61801828971)-2 = 0.26902095463

n=-3;    F-3 = (1.41986494682)-3+1 * (1.61801828971)-3 = 0.117099663048

n=-4;    F-4 = (1.41986494682)-4+1 * (1.61801828971)-4 = 0.050971237928

n=-5;    F-5 = (1.41986494682)-5+1 * (1.61801828971)-5 = 0.022186802492

n=-6;    F-6 = (1.41986494682)-6+1 * (1.61801828971)-6 = 0.009657489691

n=-7;    F-7 = (1.41986494682)-7+1 * (1.61801828971)-7 = 0.004203720079

n=-8;    F-8 = (1.41986494682)-8+1 * (1.61801828971)-8 = 0.001829798744

n=-9;    F-9 = (1.41986494682)-9+1 * (1.61801828971)-9 = 0.000796476307

n=-10;  F-10 = (1.41986494682)-10+1 * (1.61801828971)-10 = 0.000346690864

n=-11;  F-11 = (1.41986494682)-11+1 * (1.61801828971)-11 = 0.000150907885

n=-12;  F-12 = (1.41986494682)-12+1 * (1.61801828971)-12 = 0.000065687309

And so on…

 

The ‘golden ratio’ of the two consecutive terms of the sequence is equal to:

GR = 2.29736745287

 

P(ater) Sequence.

Three consecutive entries are taken into account, S, F, and P.

Pn+1 = (S)n+1 * (F)n    n >= 0       Eqn. 3

And

Pn = (S)n+1 * (F)n    n =< 0       Eqn.  4

 

‘golden ratio’, P = 3.71718255693

S = 1.61801828971

F = 2.29736745287

 

And so on…

n=10;  P10+1 = P11 = (1.61801828971)10+1 * (2.29736745287)10 = 814933.801747

n=9;    P9+1 =   P10 = (1.61801828971)9+1 * (2.29736745287)9 = 219234.269307

n=8;    P8+1 =   P9 = (1.61801828971)8+1 * (2.29736745287)8 = 58978.6124165

n=7;    P7+1 =   P8 = (1.61801828971)7+1 * (2.29736745287)7 = 15866.4826151

n=6;    P6+1 =   P7 = (1.61801828971)6+1 * (2.29736745287)6 = 4268.41629973

n=5;    P5+1 =   P6 = (1.61801828971)5+1 * (2.29736745287)5 = 1148.29342771

n=4;    P4+1 =   P5 = (1.61801828971)4+1 * (2.29736745287)4 = 308.914994119

n=3;    P3+1 =   P4 = (1.61801828971)3+1 * (2.29736745287)3 = 83.1046066176

n=2;    P2+1 =   P3 = (1.61801828971)2+1 * (2.29736745287)2 = 22.3568806064

n=1;    P1+1 =   P2 = (1.61801828971)1+1 * (2.29736745287)1 = 6.0144693633

n=0;    P0+1 =   P1 = (1.61801828971)0+1 * (2.29736745287)0 = 1.61801828971

 

n=0;    P0 = (1.61801828971)0+1 * (2.29736745287)0  = 1.61801828971

n=-1;    P-1 = (1.61801828971)-1+1 * (2.29736745287)-1  = 0.435280824907

n=-2;    P-2 = (1.61801828971)-2+1 * (2.29736745287)-2  = 0.117099663049

n=-3;    P-3 = (1.61801828971)-3+1 * (2.29736745287)-3  = 0.031502263140

n=-4;    P-4 = (1.61801828971)-4+1 * (2.29736745287)-4  = 0.008474769903

n=-5;    P-5 = (1.61801828971)-5+1 * (2.29736745287)-5  = 0.002279890421

n=-6;    P-6 = (1.61801828971)-6+1 * (2.29736745287)-6  = 0.000613338297

n=-7;    P-7 = (1.61801828971)-7+1 * (2.29736745287)-7  = 0.000165000854

n=-8;    P-8 = (1.61801828971)-8+1 * (2.29736745287)-8  = 0.000044388687

n=-9;    P-9 = (1.61801828971)-9+1 * (2.29736745287)-9  = 0.000011941487

n=-10;    P-10 = (1.61801828971)-10+1 * (2.29736745287)-10  = 0.000003212510

And so on…

 

The ‘golden ratio’ of the two consecutive terms of the sequence is equal to:

GR = 3.71718255693

 

Interesting property.

For each sequence, the product of the terms with subscript ‘n’ and ‘-n’ is equal to the square of the golden ratio of the sequence two below, that is, for example:

STn * ST-n = F2

Pn * P-n = S2

Fn * F-n = PV2

Sn * S-n = FV2

And so on…

 

In numerical values:

Pn * P-n = S2

(6.0144693633) * (0.435280824907) = 2.61798318584 = (1.61801828971)2 = (Fibonacci golden ratio) squared.

 

The next article will be about the geometric representation of the Partial Transcendental Number System, that is, the spirals.

 

 

Michele_Cavaro,_trittico_della_consolazione,_1535-40,_da_s._francesco_di_stampace_a_cagliari,_michele_arcangelo:

 

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