This one' s not full period, but here's a Golly visualization of a shift register sequence. --rwg # Pseudorandom LWSS generator with p46 logic # A lightweight spaceship stream is emitted representing a # pseudorandom binary sequence satisfying the recursion # a[n] = a[n-1] XOR a[n-19]. (By spreading out the pattern, # the 19 can be changed.) The sequence has period 413385, # so the gun has period 46*413385 = 19015710. # Built by Bill Gosper, October 1989 or earlier # (Header by Dean Hickerson, 4/11/93) x = 167, y = 139, rule = B3/S23 77b2o7b2o$77b2o6bo2b2o$76bo2bo4b6o$77b2obo5b4o$77b2obo2$77b2obo$77b2ob o5b4o$74b2ob3o4b6o11b2o$74bobobo6bo2b2o11b2o$75b4o7b2o$76b2o40$22b2o$ 22b2o5$157b2o$157b2o5$23b2o3b2o5$21b2obo3bob2o$21bo2bo3bo2bo$22b3o3b3o 126b2o5b2o$156bobo5bobo$156bo2bo3bo2bo$157b2o5b2o5$22b2o$22b2o5$157b2o $157b2o9$2b2o$2b2o6$2b3o3b3o$bo2bo3bo2bo$bo3bobo3bo$2obobobobob2o$2ob 2o3b2ob2o69b2o$b3o5b3o69bo2b2o$49b2o30bo3bo$48bo2bo30bo2bo$48bo2bo30b 3o$49b3o$82b3o$9bo72bo2bo$8bobo58b2o10bo3bo10b2o$7bo3bo37b3o17b2o10bo 2b2o10b2o$7b5o24b2o10bo2bo11b2o17b2o$6bobobobo23b2o10bo2bo11b2o$7bo3bo 37b2o2$7bo3bo$6bobobobo$2b2o3b5o76b2o$2b2o3bo3bo75bobo$8bobo75b2obo$9b o38b3o3bo32b2o26b2o$46b5o2bobo32bo25b2ob2o$41bo3b2o3b2o3bo59bo2bo$36bo 3bo4b3o3bo2bo33bo26bo2bo$35bo8bob2o3b3o33b2o27b2o$35bo2b2o5bo32b2o6b2o bo$36b2o5bo2b2o3b3o24b2o7bobo26b2o$37b3o3b5o3bo2bo33b2o25bo2bo$45b2o3b 2o3bo12b2o45bo2bo6b2o$37b3o3b8o2bobo12b2o44b2ob2o6b2o$36b2o5bo2bob3o3b o60b2o$21b2o12bo2b2o5bo$21b2o12bo8b2o$36bo3bo$41bo! Subject: Re: [math-fun] Infinitely long Gray-like codes ? Date: 2018-03-28 17:36 From: Fred Lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com> de Bruijn cycles might be a better starting point --- see OEIS A080679, and https://en.wikipedia.org/wiki/De_Bruijn_sequence WFL On 3/29/18, Henry Baker <hbaker1@pipeline.com> wrote:
Most of the applications in the Wikipedia article on Gray codes consider *fixed length* codes for some length n.
One of the properties of binary gray codes is that substrings of them contain all of the n-bit binary integers, but far more efficiently than simply concatenating all 2^n of them.
I'm interested in a slightly different *infinite* sequence of bits, which contain these n-bit substrings as early as possible, s.t., for any n, there is a function f(n) that tells my how large of an initial substring of this infinite sequence I have to generate in order to make sure that all 2^n binary integers appear at least once.
Furthermore, I may want to "tune" this infinite sequence in order to change the statistics of how frequently all of the different k-bit integers appear, how frequently all of the (k+1)-bit integers appear, etc.
Has anyone studied such sequences?