Purpose
This group is for people interested in Viswanath’s constant and the golden section.
The main focus is on new discoveries, and connections to Fibonacci numbers.


Now, Devlin adds, "mathematics has a new constant." No one has yet identified any link between this particular number and other known constants, such as the golden ratio.

Intoduction
Viswanath’s constant is equal to e..p.., and the golden ratio is equal to e..arcsinh(1/2)...
The base e, is equal to {2.718281828459045235360287471352...}.
The exponent p is between{.1239755980..., and .1239755995...}.

All possible results after the nth flip
n
1 0 1
2 00 01 10 11
3 000 001 010 011 100 101 110 111
4 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
After n flips, there is a 1/2^n chance of all heads.


The average values of all 2^n combinations converges slowly.
n a=1,b=1
0 0.00000000000000
1 1.00000000000000
2 1.18301270189222
3 1.18301270189222
4 0.876492304704711875
5 1.1435269298433128125
6 1.13574864146926828125
7 0.94521831391741234375
8 1.1369523933866469140625
9 1.1339606394862064453125
10 0.99650745428329201171875
11 1.1347508517241369921875
12 1.13330214225502626953125
13 1.0324898484942844763183593750
14 1.1337784961804310217285156250
15 1.1329850272806750823974609375
16 1.0579901750111301481628417969
17 1.1332735702819259838867187500
18 1.1328033650994049555206298828
19 1.0763597820195323508071899414
20 1.1329804183125442044067382812
21 1.1326857608493912194633483886
22 1.0897933988237643815302848816
23 1.1327952789959424537932872772
24 1.1326025999693479394119977951
25 1.0997443755758094775012135506
26 1.1326702837779045079858601093
27 1.1325399060406215252767503262
28 1.1071958332254714703949540854
29 1.1325811739163376698565110564
30 1.1324903866538227338013611734
31 1.1128269407475834475868521258


The actual number of unique approximations is much smaller than 2^n though,
because many of the sequences end up with the same integer.


There are some interesting fractalized patterns in the end terms bound by Lucas/Fibonacci, at the 3nth flip.
This could eliminate the need to calculate some of the total 2^n.
n
1 0 2 0 2
2 1 3 1 1 1 3
3 1 3 5 1 1 1 1 3 1 1 5
4 0 2 4 8 2 0 0 2 2 0 0 2 4 2 2 0 4 2 2 8
5 1 3 5 7 13 3 1 1 1 1 1 1 3 3 1 1 1 1 1 1 3 7 1 1 5 3 1 1 1 5 3 3 1 7 3 3 13
6 1 3 5 7 9 11 21 5 1 1 3 1 1 1 1 1 1 1 1 3 1 1 5 5 1 1 3 1 1 1 1 1 1 1 1 3 1 1 5 11 3 3 5 1 3 3 7 5 1 1 3 1 1 1 1 9 1 1 7 5 1 1 3 9 5 5 1 11 5 5 21


The sequences of flips that produce the closest value Viswanath's constant, out of all 2^n results, create Viswanath numbers.
The appear to alternate above or below Fibonacci numbers, + + - - + + - - ...


Vn 2 1 1 2 3 3 2 3 3 4 5 5 6 5 7 8 9 9 10 11 13 16 17 19 22 25 29 32 37 41 46 53...


Alot of terms seem to be the sum of previous terms.
3nth flip 2 2 2 4 6 8 10 16 22 32 46 68 100 144 212
3nth +1 flip 1 3 3 5 5 9 11 17 25 37 53 77 113
3nth +2 flip 1 3 3 5 7 9 13 19 29 41 59 85 125
-2 + + + -2 + -2 + + + -2


nth flip ith iteration/Viswanath's closest value
1 0 i=0 0 0.000000000000
2 00 i=0 0 1 1.000000000000
3 000 i=0 0 1 -1 1.0000000000000
4 0000 i=0 0 1 -1 2 1.18920711500272
5 00000 i=0 0 1 -1 2 -3 1.24573093961552
6 000011 i=3 0 1 -1 2 1 3 1.200936955176
7 0000001 i=1 0 1 -1 2 -3 5 2 1.10408951367381
8 00000001 i=1 0 1 -1 2 -3 5 -8 -3 1.14720269043988
9 000001000 i=8 0 1 -1 2 -3 -1 -2 1 -3 1.12983096390975
10 0000001000 i=8 0 1 -1 2 -3 5 2 3 -1 4 1.14869835499704
11 00000010000 i=16 0 1 -1 2 -3 5 2 3 -1 4 -5 1.15755791177065
12 000000000100 i=4 0 1 -1 2 -3 5 -8 13 -21 -8 -13 5 1.14352983608292
13 0000001000011 i=67 0 1 -1 2 -3 5 2 3 -1 4 -5 -1 -6 1.1477777154348
14 00000001001111i=79 0 1 -1 2 -3 5 -8 -3 -5 2 -3 -1 -4 -5 1.121828396254
15 000000010011100i=156 0 1 -1 2 -3 5 -8 -3 -5 2 -3 -1 -4 3 -7 1.13851791642936
16 0000000000010011 i=19 0 1 -1 2 -3 5 -8 13 -21 34 -55 -21 -34 13 -21 -8 1.13878863475669
17 00000010000111000 i=1080 0 1 -1 2 -3 5 2 3 -1 4 -5 -1 -6 -7 1 -8 9 1.13797288046881
18 000000000100110000 i=304 0 1 -1 2 -3 5 -8 13 -21 -8 -13 5 -8 -3 -5 2 -7 9 1.12983096390975
19 0000000010011000111 i=1223 0 1 -1 2 -3 5 -8 13 5 8 -3 5 2 3 -1 4 3 7 10 1.12883789168469
20 00000000010011000011 i=1219 0 1 -1 2 -3 5 -8 13 -21 -8 -13 5 -8 -3 -5 2 -7 9 2 11 1.12737820415783
21 000000000100110000111 i=2439 0 1 -1 2 -3 5 -8 13 -21 -8 -13 5 -8 -3 -5 2 -7 9 2 11 13 1.12991278190825
22 0000000000100110011111 i=2463 0 1 -1 2 -3 5 -8 13 -21 34 13 21 -8 13 5 8 -3 5 2 7 9 16 1.13431252219546
23 00000000001001100111000 i=4920 0 1 -1 2 -3 5 -8 13 -21 34 13 21 -8 13 5 8 -3 5 2 7 -5 12 -17 1.1310916053653
24 000000000010011001110011 i=9843 0 1 -1 2 -3 5 -8 13 -21 34 13 21 -8 13 5 8 -3 5 2 7 -5 12 7 19 1.13052820036829
25 0000000000100110011100011 i=19683 0 1 -1 2 -3 5 -8 13 -21 34 13 21 -8 13 5 8 -3 5 2 7 -5 12 -17 -5 -22 1.13161034025618
26 00000000000010011001100000 i=9824 0 1 -1 2 -3 5 -8 13 -21 34 -55 89 34 55 -21 34 13 21 -8 13 5 8 -3 11 -14 25 1.13179279115279


Golden string property
Each power of Viswanath’s constant(1.131988248...)..n.. has a corresponding coin flip.
Simply evaluate the absolute values of the two possible(+Heads or -Tails) outcomes, and choose the flip that produces the closest value to that of the power.
All(almost surely) random sequences will generally follow the powers of the constant, but there is only one that is closest.
Below is a binary notation for heads/tails, so that heads is 1, and tails is 0.

GS = 10110101101101011010110110101101... Golden string
VS = 10100101101001011010010110100101... Viswanath’s string

The gold string, otherwise known as the rabbit constant when in decimal form, is remarkably similar to the sequence above,
Continued fractions
It seems that it may have a "simple" continued fraction with a pattern, and may even have a "non-simple" continued fraction.
Upper bound: e...1239755995..... = 1.131988249582904392... with the continued fraction = [1; 7 1 1 2 1 3 2 1 2 1 8 2 5 3 5 1 11...]
Middle bound: e...1239755988..... = 1.131988248790512617... with the continued fraction = [1; 7 1 1 2 1 3 2 1 2 1 8 1 5 1 2 1 10...]
Lower bound: e...1239755980..... = 1.131988247884922019... with the continued fraction = [1; 7 1 1 2 1 3 2 1 2 1 9 2 1 7 1 3 3...]
Here are the convergent integers for each CF term:
1: 1/1 = 1.000000000000000...
7: 8/7 = 1.1428571428571428...
1: 9/8 = 1.1250000000000000...
1: 17/15 = 1.1333333333333333...
2: 43/38 = 1.131578947368421...
1: 60/53 = 1.1320754716981131...
3: 223/197 = 1.131979695431472...
2: 506/447 = 1.1319910514541387...
1: 729/644 = 1.1319875776397516...
2: 1964/1735 = 1.1319884726224784...
1: 2693/2379 = 1.1319882303488861...

Convergents
Here is an easy Fibonacci-like way to recursively generate the convergents
for a simple continued fraction.

Numerators:
1+ 1(8) = 9
8 + 1(9) = 17
9 + 2(17) = 43
17 + 1(43) = 60
43 + 3(60) = 223
60 + 2(223) = 506
223 + 1(506) = 729
506 + 2(729) = 1964
729 + 1(1964) = 2693

Denominators:
1+ 1(7) = 8
7 + 1(8) = 15
8 + 2(15) = 38
15 + 1(38) = 53
38 + 3(53) = 197
53 + 2(197) = 447
197 + 1(447) = 644
447 + 2(644) = 1735
644 + 1(1735) = 2379

The multipliers outside the parenthesis are actually the terms of the continued fraction.