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.

Quote:
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...}.

Observation
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 one unique path, that follows them the 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 almost identical to the sequence above,
and is also embedded within the bits that don’t match.
If you use the same method as above with the powers of V³, the string of heads and tails matches the pattern of the golden string.
So the golden string, may be used to obtain Viswanath’s constant raised to the 3^rdpower.

Download
Viswanath.exe attached to this page.

Algorithm
The first seven bits are zero based, with the golden string’s zero^th bit.
For even bits of the gold string write zeroes, and for odd bits write ones.
For each bit of the golden string that is a 1, write seven consecutive bits, and for each 0 of the golden string write eight consecutive bits.
You may start the first Vibonacci terms as: a=0, b=1 or a=0, b=-1 which just reverses the sign and produces the same sequence.
For example:

V³ =
000000011111110000000011111110000000111111110000000111111110000000111111100000000111111100000001111111100000001111111100000001111111000000001111111000000001111111000000011111111...

This finite set of flips creates the average value: 1.13155109981581...
With the terms:
HT i (V³)ⁱ Terms
0 1 1 -1
0 2 2 2
0 3 3 -3
0 4 4 5
0 5 6 -8
0 6 9 13
0 7 14 -21
1 8 20 -8
1 9 28 -29
1 10 41 -37
1 11 60 -66
1 12 87 -103
1 13 126 -169
1 14 183 -272
0 15 265 103
0 16 384 -375
0 17 557 478
0 18 808 -853
0 19 1172 1331
0 20 1700 -2184
0 21 2466 3515
0 22 3577 -5699
1 23 5189 -2184
1 24 7527 -7883
1 25 10918 -10067
1 26 15837 -17950
1 27 22972 -28017
1 28 33321 -45967
1 29 48333 -73984
0 30 70109 28017
0 31 101695 -102001
0 32 147511 130018
0 33 213969 -232019
0 34 310367 362037
0 35 450196 -594056
0 36 653021 956093
1 37 947225 362037
1 38 1373975 1318130
1 39 1992987 1680167
1 40 2890881 2998297
1 41 4193300 4678464
1 42 6082494 7676761
1 43 8822821 12355225
1 44 12797739 20031986
0 45 18563463 -7676761
0 46 26926800 27708747
0 47 39058045 -35385508
0 48 56654741 63094255
0 49 82179219 -98479763
0 50 119203160 161574018
0 51 172907377 -260053781
1 52 250806783 -98479763
1 53 363801959 -358533544
1 54 527704489 -457013307
1 55 765449501 -815546851
1 56 1110305010 -1272560158
1 57 1610527166 -2088107009
1 58 2336112806 -3360667167
1 59 3388594219 -5448774176
0 60 4915246707 2088107009
0 61 7129697046 -7536881185
0 62 10341816595 9624988194
0 63 15001082066 -17161869379
0 64 21759471471 26786857573
0 65 31562696383 -43948726952
0 66 45782536783 70735584525
1 67 66408796283 26786857573
1 68 96327738341 97522442098
1 69 139725965433 124309299671
1 70 202676256625 221831741769
1 71 293987340666 346141041440
1 72 426436514623 567972783209
1 73 618557590240 914113824649
0 74 897234358042 -346141041440
0 75 1301462476499 1260254866089
0 76 1887806193057 -1606395907529
0 77 2738313464197 2866650773618
0 78 3971997049157 -4473046681147
0 79 5761488143996 7339697454765
0 80 8357192924009 -11812744135912
0 81 12122332255755 19152441590677
1 82 17583767738177 7339697454765
1 83 25505726237076 26492139045442
1 84 36996739297701 33831836500207
1 85 53664761628014 60323975545649
1 86 77842174614845 94155812045856
1 87 112912159952740 154479787591505
1 88 163782113337336 248635599637361
0 89 237570343712064 -94155812045856
0 90 344602148924660 342791411683217
0 91 499854650155407 -436947223729073
0 92 725052562967648 779738635412290
0 93 1051708169369862 -1216685859141363
0 94 1525530878743560 1996424494553653
0 95 2212823414117323 -3213110353695016
1 96 3209759651734297 -1216685859141363
1 97 4655842375931783 -4429796212836379
1 98 6753424113176095 -5646482071977742
1 99 9796022624864005 -10076278284814121
1 100 14209393288305876 -15722760356791863
1 101 20611105685819614 -25799038641605984
1 102 29896961043484130 -41521798998397847
1 103 43366343041487538 -67320837640003831
0 104 62904042523139628 25799038641605984
0 105 91243999107037217 -93119876281609815
0 106 132351865462103417 118918914923215799
0 107 191979927038815192 -212038791204825614
0 108 278472028007659194 330957706128041413
0 109 403931137899744393 -542996497332867027
0 110 585912938302351504 873954203460908440
1 111 849882415738151481 330957706128041413
1 112 1232777215457536298 1204911909588949853
1 113 1788176381589554159 1535869615716991266
1 114 2593797753220117866 2740781525305941119
1 115 3762373138285852759 4276651141022932385
1 116 5457423044692436440 7017432666328873504
1 117 7916138350463954729 11294083807351805889
1 118 11482570779377394318 18311516473680679393
0 119 16655776575163577637 -7017432666328873504
0 120 24159650190879092606 25328949140009552897
0 121 35044219926437847592 -32346381806338426401
0 122 50832579964927751915 57675330946347979298
0 123 73734019228129701509 -90021712752686405699
0 124 106953170491942149544 147697043699034384997
0 125 155138439461529406877 -237718756451720790696
1 126 225032463159863995055 -90021712752686405699
1 127 326415617248444403299 -327740469204407196395
1 128 473474598675975983725 -417762181957093602094
1 129 686787591479570615648 -745502651161500798489
1 130 996203802969171603157 -1163264833118594400583
1 131 1445020308116269966013 -1908767484280095199072
1 132 2096040674252532804190 -3072032317398689599655
0 133 3040363158527671995292 1163264833118594400583
0 134 4410128223789735439586 -4235297150517284000238
0 135 6397009151921608783083 5398561983635878400821
0 136 9279033173916140600468 -9633859134153162401059
0 137 13459486237685368795426 15032421117789040801880
0 138 19523345416167499082976 -24666280251942203202939
0 139 28319135627314765394281 39698701369731244004819
0 140 41077664999673946233557 -64364981621673447207758
1 141 59584253701511568530013 -24666280251942203202939
1 142 86428556472094395139613 -89031261873615650410697
1 143 125366936896962581937583 -113697542125557853613636
1 144 181848101003533239203012 -202728803999173504024333
1 145 263775542875151983371610 -316426346124731357637969
1 146 382613492442955328234265 -519155150123904861662302
1 147 554991122390315674355152 -835581496248636219300271
0 148 805029493250254355433238 316426346124731357637969
0 149 1167716849616529781533336 -1152007842373367576938240
0 150 1693804577734236199173500 1468434188498098934576209
0 151 2456908923165410861868060 -2620442030871466511514449
0 152 3563812222543745572337233 4088876219369565446090658
0 153 5169405116242119554295146 -6709318250241031957605107
0 154 7498360628202874261875015 10798194469610597403695765
0 155 10876572999458758637656351 -17507512719851629361300872
1 156 15776760558515324447383328 -6709318250241031957605107
1 157 22884613906706723873773549 -24216830970092661318905979
1 158 33194745633403858297045986 -30926149220333693276511086
1 159 48149867948764328887479185 -55142980190426354595417065
1 160 69842673569109311978494172 -86069129410760047871928151
1 161 101308669350283123166641616 -141212109601186402467345216
1 162 146950939318915344415125156 -227281239011946450339273367
0 163 213156274830207216306801022 86069129410760047871928151
0 164 309188887870160188473119361 -313350368422706498211201518
0 165 448486766146276043724932449 399419497833466546083129669
0 166 650542070880667469141996674 -712769866256173044294331187
0 167 943628704191635175018139771 1112189364089639590377460856
0 168 1368758718662058616889953702 -1824959230345812634671792043
0 169 1985421195424894571200581036 2937148594435452225049252899
1 170 2879906640591530751106751250 1112189364089639590377460856
1 171 4177381745311855326842893701 4049337958525091815426713755
1 172 6059404148767960976882987551 5161527322614731405804174611
1 173 8789328071180475507123223223 9210865281139823221230888366
1 174 12749155865193206408013558850 14372392603754554627035062977
1 175 18492992178543155816529794166 23583257884894377848265951343
1 176 26824580649244078823066920330 37955650488648932475301014320
1 177 38909772959439242122975263346 61538908373543310323566965663

Primes
There seems to be a primal pattern similar to that of the Fibonacci primes.
If we use 1 as the starting values for both a and b, then list only the new numbers of the sequence, we get:
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 15
10 22
11 29
12 36
13 43
14 50
15 57
16 107
17 157
18 207
19 257
20 307
21 357
22 407
23 764
24 1121
25 1478
26 2599
27 3720
28 4841
29 8561
30 12281
31 20842
32 33123
33 45404
34 57685
35 103089
36 160774
37 218459
38 276144
39 494603
40 713062
41 1207665
42 1920727
43 3128392
44 5049119
45 8177511
46 13226630
47 21404141
48 34630771
49 56034912
50 77439053
51 98843194
52 176282247
53 253721300
54 331160353
55 584881653
56 916042006
57 1500923659
58 2416965665
59 3917889324
60 6334854989
61 10252744313
62 16587599302
63 26840343615
64 43427942917
65 70268286532
66 113696229449
67 183964515981
68 297660745430
69 481625261411
70 665589777392
71 849554293373
72 1515144070765
73 2180733848157
74 2846323625549
75 5027057473706
76 7873381099255
77 12900438572961
78 20773819672216
79 33674258245177
80 54448077917393
81 88122336162570
82 142570414079963
83 230692750242533
84 373263164322496
85 603955914565029
86 977219078887525
87 1581174993452554
88 2558394072340079
89 4139569065792633
90 6697963138132712
91 9256357210472791
92 11814751282812870
93 21071108493285661
94 30327465703758452
95 39583822914231243
96 69911288617989695
97 109495111532220938
98 179406400150210633
99 288901511682431571
100 468307911832642204
101 757209423515073775
102 1225517335347715979
103 1982726758862789754
104 3208244094210505733
105 5190970853073295487
106 8399214947283801220
107 13590185800357096707
108 21989400747640897927
109 35579586547997994634
110 57568987295638892561
111 93148573843636887195
112 128728160391634881829
113 164307746939632876463
114 293035907331267758292
115 421764067722902640121
116 550492228114537521950
117 972256295837440162071
118 1522748523951977684021
119 2495004819789417846092
120 4017753343741395530113
121 6512758163530813376205
122 10530511507272208906318
123 17043269670803022282523
124 27573781178075231188841
125 44617050848878253471364
126 72190832026953484660205
127 116807882875831738131569
128 188998714902785222791774
129 305806597778616960923343
130 494805312681402183715117
131 800611910460019144638460
132 1106418508238636105561803
133 1412225106017253066485146
134 2518643614255889172046949
135 3625062122494525277608752
136 4731480630733161383170555
137 8356542753227686660779307
138 13088023383960848043949862
139 21444566137188534704729169
140 34532589521149382748679031
141 55977155658337917453408200
142 90509745179487300202087231
143 146486900837825217655495431
144 236996646017312517857582662
145 383483546855137735513078093
146 620480192872450253370660755
147 1003963739727587988883738848
148 1624443932600038242254399603
149 2628407672327626231138138451
150 4252851604927664473392538054
151 6881259277255290704530676505
152 11134110882182955177923214559
153 15386962487110619651315752613
154 19639814092038284124708290667
155 35026776579148903776024043280
156 50413739066259523427339795893
...(In progress)

So far all of the indexes that produce a prime, are either prime themselves or an even number that is one more or less than a prime, although this may not continue to hold.

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...

The next term in the CF, is either an eight or a nine.
8.4570118469086157043553123078555...: 23508/20767 = 1.1319882505898781...
9.3465054755990290883410593840048...: 26201/23146 = 1.1319882485094616...

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.

Solve
Use these two formula to solve for p:
arccosh(x) = ln[x+(x²-1)^(1/2)]
arcsinh(x) = ln[x+(x²+1)^(1/2)]

Then V = e^arccosh(a) and V = e^arcsinh(b)
Then V = (a + b) and V = 1 / (a - b)
Then a = Sqrt(b²+1) and b = Sqrt(a²-1)
Then a = Sqrt(1.015448855...) and b = Sqrt(0.015448855...)
Then a = 1.0076948.... and b = 0.124293426...













Links
The random Fibonacci recurrence and the visible
points of the plane