for example,
fibonacci numbers: Fi+1 = Fi + Fi-1 -> phi as i -> inf. (golden ratio)
padovan numbers: Pi+1 = Pi-1 + Pi-2 -> rho as i -> inf. (plastic ratio)
definition:
let Fi(n,m) be an n-order fibonacci-type number where n is the number of elements on the
left to skip and m is the number of elements to add, and i is the ith element in the list
for example:
fibonacci: Fi(0,2) = Fi+1 = Fi + Fi-1, i.e. skip 0 numbers to the left add two
tribonacci: Fi(0,3) = Ti+1 = Ti + Ti-1 + Ti-2i, i.e. skip zero numbers to the left add three
...
padovan: Fi(1,2) = Pi+1 = Pi-1 + Pi-2, i.e. skip one number and add the next two
"tri"-adovan: Fi(1,3) = Pi+1 = Pi-1 + Pi-2 + Pi-3, i.e. skip one number and add the next three
...
converegence of Fi(n,m) series:
let F(n,m) be the converegnce limit of an Fi(n,m) series defined as:
for (n>=0 and m>=2) define F(n,m) = lim i -> inf. Fi+1(n,m)/Fi(n,m)
examples:
F(0,2) = lim i-> inf. Fi+1(0,2)/Fi(0,2) = phi (golden ratio)
F(1,2) = lim i-> inf. Fi+1(1,2)/Fi(1,2) = rho (plastic ratio)
convergence of F(n,m) limits:
let F(n) be the convergence limit of an F(n,m) series defined as:
(for n>=0) define F(n) = lim m -> inf. F(n,m) = lim m,i -> inf. Fi+1(n,m)/Fi(n,m)
examples:
F(0) = lim m,i->inf. Fi+1(0,m)/Fi(0,m) = 2 (fibonacci sequeunce)
F(1) = lim m,i->inf. Fi+1(1,m)/Fi(1,m) = 1.6180339887498952 (padovan sequence)
F(2) = lim m,i->inf. Fi+1(2,m)/Fi(2,m) = 1.465571231876768
F(3) = lim m,i->inf. Fi+1(3,m)/Fi(3,m) = 1.3802775690976141
Some properties of the convergences:
fibonnaci constant F(0,2)
phi ~ 1.618033988749895
phi^2 ~ 2.618033988749895
phi^3 ~ 4.236067977499790 = phi^2 + phi
phi^4 ~ 6.854101966249685 = phi^3 + phi^2
plastic ratio F(1,2)
rho ~ 1.3247179572447460
rho^2 ~ 1.75487766624669
rho^3 ~ 2.32471795724475
rho^4 ~ 3.07959562349144 = rho^2 + rho
rho^5 ~ 4.07959562349144 = rho^3 + rho^2
F(2,2)
F(2,2) ~ 1.220744085
F(2,2)^2 ~ 1.49021612
F(2,2)^3 ~ 1.819172513
F(2,2)^4 ~ 2.220744085
F(2,2)^5 ~ 2.710960205 = F(2,2)^2 + F(2,2)
Conject that for a given F(n,2)
F(n,2)^(n+3) = F(n,2)^2 + F(n,2)
Easy way to compute limits
F(1) = phi = F(0,2)
F(2) = 1.465571231876768 = F(1,3)
F(3) = 1.3802775690976141 = F(2,4)
F(4) = rho = F(3,5)
conject the limit of the series F(n) is given by an element in the previous series:
F(n) = F(n-1,n+1)
First 100 series limits:
F(0) ~ 2.0
F(1) ~ 1.618033988749895 x^2 = x + 1
F(2) ~ 1.465571231876768 x^3 = x^2 + 1
F(3) ~ 1.3802775690976141
F(4) ~ 1.324717957244746 x^3 = x + 1
F(5) ~ 1.2851990332453493
F(6) ~ 1.2554228710768465
F(7) ~ 1.2320546314285723
F(8) ~ 1.21314972305964
F(9) ~ 1.1974914335516809
F(10) ~ 1.1842763223508939
F(11) ~ 1.1729507500239802
F(12) ~ 1.1631197906692043
F(13) ~ 1.1544935507090563
F(14) ~ 1.1468540421995068
F(15) ~ 1.140033937477005
F(16) ~ 1.1339024903348374
F(17) ~ 1.128355939691603
F(18) ~ 1.1233108062463268
F(19) ~ 1.118699108052226
F(20) ~ 1.1144648799534382
F(21) ~ 1.1105615981223105
F(22) ~ 1.1069502450168822
F(23) ~ 1.1035978353382538
F(24) ~ 1.1004762790343705
F(25) ~ 1.0975614942323275
F(26) ~ 1.0948327079053122
F(27) ~ 1.092271899234358
F(28) ~ 1.0898633526169947
F(29) ~ 1.0875932957790584
F(30) ~ 1.0854496045569988
F(31) ~ 1.0834215603634179
F(32) ~ 1.0814996496192069
F(33) ~ 1.0796753968675168
F(34) ~ 1.0779412251108815
F(35) ~ 1.0762903382967246
F(36) ~ 1.0747166219343687
F(37) ~ 1.0732145586419175
F(38) ~ 1.071779156054469
F(39) ~ 1.0704058850202818
F(40) ~ 1.069090626401449
F(41) ~ 1.067829625104681
F(42) ~ 1.066619450214228
F(43) ~ 1.0654569602966095
F(44) ~ 1.0643392731061976
F(45) ~ 1.0632637390499107
F(46) ~ 1.0622279178745109
F(47) ~ 1.0612295581261757
F(48) ~ 1.0602665790028432
F(49) ~ 1.0593370542783376
F(50) ~ 1.0584391980257963
F(51) ~ 1.0575713519083083
F(52) ~ 1.0567319738384184
F(53) ~ 1.055919627836474
F(54) ~ 1.055132974941605
F(55) ~ 1.0543707650492578
F(56) ~ 1.053631829566251
F(57) ~ 1.0529150747888052
F(58) ~ 1.0522194759213541
F(59) ~ 1.0515440716645008
F(60) ~ 1.0508879593095317
F(61) ~ 1.0502502902846866
F(62) ~ 1.0496302661050787
F(63) ~ 1.049027134683961
F(64) ~ 1.048440186968044
F(65) ~ 1.04786875386393
F(66) ~ 1.0473122034265143
F(67) ~ 1.04676993828351
F(68) ~ 1.0462413932731371
F(69) ~ 1.0457260332745495
F(70) ~ 1.0452233512127838
F(71) ~ 1.0447328662219708
F(72) ~ 1.0442541219522632
F(73) ~ 1.0437866850074473
F(74) ~ 1.0433301435015525
F(75) ~ 1.0428841057239469
F(76) ~ 1.0424481989034702
F(77) ~ 1.0420220680630814
F(78) ~ 1.0416053749573362
F(79) ~ 1.0411977970857405
F(80) ~ 1.040799026775703
F(81) ~ 1.0404087703293814
F(82) ~ 1.0400267472292637
F(83) ~ 1.0396526893977895
F(84) ~ 1.0392863405067418
F(85) ~ 1.0389274553325312
F(86) ~ 1.0385757991538263
F(87) ~ 1.038231147188305
F(88) ~ 1.0378932840655741
F(89) ~ 1.0375620033335595
F(90) ~ 1.0372371069959012
F(91) ~ 1.0369184050780864
F(92) ~ 1.036605715220252
F(93) ~ 1.0362988622947453
F(94) ~ 1.0359976780467006
F(95) ~ 1.0357020007560167
F(96) ~ 1.0354116749192566
F(97) ~ 1.0351265509501044
F(98) ~ 1.0348464848971208
F(99) ~ 1.0345713381776367
also see: http://www.mscroggs.co.uk/blog/tags/golden%20spiral