I understand how your pattern could be seen as being "sub-linear", but in the current jargon used, we usually mean that the population N relates to its time T in a linear fashion, so that the ratio of the two value close to some unknown constant that could be higher or lower than 1, both. In you more constrained definition of linear, the number of "truly" linear pattern would be very rare, I can only think of the following (though it is N = T + 44):
Code: Select all
x = 17, y = 15, rule = B3/S23
8b2o$7b2o$9bo$11b2o$10bo2$9bo2b2o$b2o5b2o4bo$2o5bo5bo$2bo4bobo3b2o$4bo
2bo4b2obo$4b2o7b2o$8bo4bob2o$7bobo2bob2o$8bo!
Having the above pasted twice is still linear (if there is no collision), and half of it (if it worked) would also be still linear. The formal way to state this is to define that population to time relation as N = a*T^1 + b, where 1 is the exponent to T, if it is 1 it is linear, if it is less, it is sublinear; however "a" can be anything (except 0). There is many types of sublinear growth, many involve logarithmic growth, and to know what is sublinear or not, one needs the notion of proving the "population function" cannot be bound from below by linear function in the limit as T goes to infinity. Such formality is definitely needed when one does not know the exact population function, which is a headache to derive, especially for such a sawtooth pattern as you described. From what I understood from your design, it should be in the same growth class as the following:
Code: Select all
x = 131, y = 75, rule = B3/S23
59bo$56b2ob2o$55bobobo$54b2o3bo$55bobo2bo$56b2obo2$60b2o$6bo2b2o48bob
3o$5bo6b2o2bo15b2o24bo3b3o44b2obo$5bo4bo3b2o15b2ob2o15bo7bobo48bob2o
14b2o$5bob2o2b3ob2o4b2o9bo3bo13bobo7bo18bobo25bo2bo3bo10bo4bo$2b2ob3o
6b2o3bo3bo8bo4bo11b2obo25bo28bo2b4o11bo3b2o$b6o9bobo7b2o5bobob2o8bobob
3o25bo2b2o23bobo13b2obo$o4b2o9b2o5b5o3bobo2bob2o2bo2b2o6bo27b3o26bobo
10bo2b2o$b2ob3o13b3o4b2o3bo5b2o3bo2b2o3bo26bo31bo12bo4bo$2b2o6bo6b2o2b
2o2b2o3bo2bo5bo3bo7bo2b2o21b2o3b3o24b4obo9bo$4b7o11bo2b2o3b3o8bo9bo3b
2o19b2o4b3o3b2o19b2o2bo13b2o$42bobo12bob3o15b2o8bo2bo19b3o$4b7o11bo2b
2o3b3o8bo9bo3b2o31b2o$2b2o6bo6b2o2b2o2b2o3bo2bo5bo3bo7bo2b2o$b2ob3o13b
3o4b2o3bo5b2o3bo2b2o3bo22b2o$o4b2o9b2o5b5o3bobo2bob2o2bo2b2o6bo20b2o$b
6o9bobo7b2o5bobob2o8bobob3o26bobo$2b2ob3o6b2o3bo3bo8bo4bo11b2obo27b2o$
5bob2o2b3ob2o4b2o9bo3bo13bobo28bo$5bo4bo3b2o15b2ob2o15bo7bo3bo$5bo6b2o
2bo15b2o25b2o2b2o$6bo2b2o46b2ob2o2bo$56b2obob2obo$55bo3bo5bo$56b3o$57b
o7bo$63bo10b2ob2o$73bo$53b7o12b2ob2o$53bo6bo12bo3bo$53b2o3bo17bo$14bob
o12b2o24bob2o13bo$13bo2bo10bo4bo25b2o10bo19bo$12b2o7bo4bo31bobo8b2ob3o
15bobo$11bo6b2obo4bo5bo24bo10b2o20b2o$10b6o2bo7b6o24bo4bo7b2obo$7b2o7b
o3b4o32bo5bo7bo$6bo3b3obo4bo36bo2b2o11b3o$5bo3bo3b2obo2b2obo2bo31b4o
12b2o$5bo5b2o3bo5bo35bo$5b3o3b4obo7bob2o$14bo9bo2bo6bo$5b3o6bobo7bo2bo
4bobo$4bo5bo5bo8b2o6b2o5bo2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o$3bo3b2o
bo3b2o23b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o$3bo3bo6b3o$3bo3b2obo3b
2o23b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o$4bo5bo5bo8b2o6b2o5bo2b2o2b
2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o$5b3o6bobo7bo2bo4bobo$14bo9bo2bo6bo$5b3o
3b4obo7bob2o$5bo5b2o3bo5bo$5bo3bo3b2obo2b2obo2bo$6bo3b3obo4bo$7b2o7bo
3b4o$10b6o2bo7b6o64b2ob2o$11bo6b2obo4bo5bo62bo$12b2o7bo4bo67b2ob2o$13b
o2bo10bo4bo62bo3bo$14bobo12b2o67bo$94bo$92bo$91b2ob3o$90b2o$91b2obo$
92bo$94b3o$95b2o!
In any case, there are useful links to several sublinear patterns in the wiki:
https://conwaylife.com/wiki/Sublinear_growth