I wrote:

Sawtooth with expansion factor 6, based on zdr's c/10 copperhead spaceship.

...

Guns of any period 48N for N>=2 could be used, giving expansion factor N+1.

In fact the period only needs to be a multiple of 16, if we modify things a bit, so we can get expansion factors that aren't integers. It takes some extra work, because when the loaf is pulled all the way back it can be in any of 3 different positions, and we have to arrange for its deletion to release another HWSS in all 3 cases. The smallest period that I was able to make work is 112, giving expansion factor 10/3:

Code: Select all

```
#C Sawtooth with expansion factor 10/3, based on zdr's c/10 copperhead
#C spaceship.
#C A p112 shotgun produces salvos consisting of a HWSS and two LWSSs.
#C Usually the HWSS is deleted by a glider, but occasionally one is
#C allowed to escape. When it does, it eventually hits the back of
#C the copperhead, which starts a loaf being pulled back toward the
#C shotgun. When the loaf reaches the shotgun, it's deleted, in one
#C of 3 different ways, and another HWSS escapes.
#C Because the e.f. is not an integer, there's no simple formula for the
#C generations when particular things happen. However, a recursive formula
#C exists: Define P(0)=245 and P(n+1) = (10 P(n) - c(P(n) mod 3))/3, where
#C c(0)=735, c(1)=1435, and c(2)=1295. Then a HWSS hits the back of the
#C copperhead in generation P(n). The loaf is destroyed in generation
#C (8*P(n)-d(P(n) mod 3))/3, where d(0)=951, d(1)=1181, and d(2)=1069.
#C The minimum repeating population is 2097. It occurs in generation
#C (8*P(n)-e(P(n) mod 3))/3, provided that P(n) mod 3 is 0 or 1, where
#C e(0)=1512 and e(1)=1736. The successive maximum populations occur
#C around generations P(n) and are approximately 3*P(n)/70.
#C Dean Hickerson, 3/8/2016
x = 280, y = 250, rule = B3/S23
206bo7bob2o$204b3o7b2obo27b2o$188bo14bo41bo$188b3o12b2o38bobo$191bo47b
2o2b2o$190b2o47b2o$210bo$211bo$102bo7bob2o77b2o18bo$100b3o7b2obo27b2o
48b2o17b2o$84bo14bo41bo$84b3o12b2o38bobo65b3o$87bo47b2o2b2o$86b2o47b2o
2$98bob3o$87b2o10bo2bo104b2o36b2o$87b2o13bo3b2o99bo19b2o16bobo$100b2o
4b2o100b3o15bobo18bo$98b2o110bo15bo20b2o$204b2o19b2o4b2o4bo$204bo25bob
o4bo$205b3o22bo6bo$133bo3bo69bo21b2o8bo2bo$103b2o28bo2bo4b2o96b3o$31b
2obo7bo60bo19b2o9bo2bo3bobo$2b2o27bob2o7b3o59b3o15bobo9bo2bo5bo$3bo41b
o14bo45bo15bo12b3o5b2o102b2o$3bobo38b2o12b3o39b2o19b2o4b2o118bo$4b2o2b
2o47bo42bo25bobo119bo$8b2o47b2o42b3o22bo120b2o$103bo21b2o7b2o$134b2o
47bo$40bo15b2o125bobo$40b2o14b2o125b2o$40bobo100b2o$143bo$144bo49b3o$
143b2o48bo2bo8b2o21bo$83b2o113bo6bo22b3o$83bo114bo4bobo25bo$2b2o36b2o
42bo113bo4b2o4b2o19b2o$bobo16b2o19bo41b2o102b2o20bo15bo$bo18bobo15b3o
147bo18bobo15b3o$2o8bo11bo15bo149bobo16b2o19bo$9bo6b2o4b2o19b2o47b2o
95b2o36b2o$8b3o5bobo25bo47b2o7b2o21bo$18bo22b3o57bo22b3o$18b2o21bo57bo
bo25bo$99b2o4b2o19b2o$83b2o5b3o12bo15bo104b3o$84bo5bo2bo9bobo15b3o$2o
82bobo3bo2bo9b2o19bo99b2o17b2o$bo83b2o4bo2bo28b2o99bo18b2o$o61bo27bo3b
o129bo$2o61bo161bo$61b3o131b2o47b2o$191b2o2b2o47bo$128b2o60bobo38b2o
12b3o$120b2o4b2o62bo41bo14bo$120b2o3bo13b2o14bo33b2o27bob2o7b3o$125bo
2bo10b2o14bobo60b2obo7bo$125b3obo25b2o$20bo21b2o$18b3o22bo47b2o47b2o
84bo7bob2o$17bo25bobo5b3o33b2o2b2o47bo83b3o7b2obo27b2o$17b2o19b2o4b2o
6bo33bobo20bobo15b2o12b3o64bo14bo17b2o22bo$23bo15bo11bo8b2o24bo23b2o
16bo14bo64b3o12b2o15b2o21bobo$21b3o15bobo18bo24b2o23bo14b3o83bo30bo16b
2o2b2o$20bo19b2o16bobo64bo84b2o8b2o37b2o$20b2o36b2o164bo$221bo3bo$211b
2o12bo$211b2o7bobob2o4b2o$222b2o6b2o2$19bobo$4b2o14b2o237bo3bo$4b2o15b
o239bobo$258b2o$227b2o29bo6b2o$3b2o47b2o173bo19b2o10bo2bo2bobo$4bo47b
2o2b2o32bo137b3o15bobo10b3o5bo$b3o12b2o38bobo32bo138bo15bo20b2o$bo14bo
41bo30b3o132b2o19b2o4b2o$17b3o7b2obo27b2o164bo25bobo$19bo7bob2o194b3o
22bo$227bo21b2o7b2o$46b2obo7bo69bo130b2o$17b2o27bob2o7b3o67bobo$18bo
41bo14bo51b2o$18bobo38b2o12b3o191b2o$19b2o2b2o47bo194bo$23b2o47b2o194b
o$53bo213b2o$52bo$52bo18b2o$52b2o17b2o192bo$265b2o$54b3o207bobo2$216b
2o$216b2o7b2o21bo$225bo22b3o$17b2o36b2o166bobo25bo$16bobo16b2o19bo166b
2o4b2o19b2o$16bo18bobo15b3o151b2o20bo15bo$15b2o20bo15bo154bo5b3o10bobo
15b3o$26bo4b2o4b2o19b2o148bobo2bo2bo10b2o19bo$26bo4bobo25bo149b2o6bo
29b2o$26bo6bo22b3o157b2o$21bo2bo8b2o21bo155bobo$22b3o187bo3bo3$15b2o
227b2o6b2o$16bo82bo144b2o4b2obobo7b2o$15bo83bobo148bo12b2o$15b2o82b2o
149bo3bo$251bo$80bo134b2o37b2o8b2o$78bobo130b2o2b2o16bo30bo$79b2o129bo
bo21b2o15b2o12b3o$210bo22b2o17bo14bo$209b2o27bob2o7b3o$67b3o168b2obo7b
o$35bo21b2o8bo2bo$33b3o22bo6bo$32bo25bobo4bo$32b2o19b2o4b2o4bo$38bo15b
o20b2o$36b3o15bobo18bo$35bo19b2o16bobo95bo$35b2o36b2o3b4o89b2ob3o$77bo
4bob2o80b2o7b3o$77bo4bo3bo81bo4b2o3bo$80b2o4bo81bo4b2o3bo$77b2o4bo2bo
79b2o7b3o$35b3o38bo2bo3bo2bo84b2ob3o$76bo2bo4b2o6bo78bo$19b2o17b2o36bo
4b2o8b3o$19b2o18bo36bo3bo4bo4b3obo$39bo37b2obo4bo5bo3bo$38bo42b4o7bo3b
o152b2obo7bo$18b2o47b2o24bob3o122b2o27bob2o7b3o$19bo47b2o2b2o21b3o124b
o41bo14bo$16b3o12b2o38bobo21bo125bobo38b2o12b3o$16bo14bo41bo148b2o2b2o
47bo$32b3o7b2obo27b2o151b2o47b2o$34bo7bob2o2$274b2o$42bo7bob2o184bo16b
2o17b2o$40b3o7b2obo27b2o153b2ob2o14b2o$24bo14bo41bo151b5o$24b3o12b2o
38bobo151bo2bo$27bo47b2o2b2o152bo2bo$26b2o47b2o157b2o$266b3o$44bo175b
2o36b2o5bo2bo$27b2o15b2o173bobo16b2o19bo5bo2bo$27b2o14bo2bo172bo18bobo
15b3o7b4o$44b2o172b2o20bo15bo11b2o$234b2o4b2o19b2o7bo$234bobo25bo6bo$
236bo22b3o7bo$227b2o7b2o21bo$227b2o$43b2o36b2o$43bo19b2o16bobo$44b3o
15bobo18bo$46bo15bo20b2o$40b2o19b2o4b2o6bo$40bo25bobo5bob2o$41b3o22bo
7bobo142b2o57b2o$43bo21b2o8bo143b2o58bo$278bo$278b2o3$269b2o$85b2o151b
o21b2o7b2o$86b2o140bo7b3o22bo$23b2o60bo142bo6bo25bobo$23bo203bo7b2o19b
2o4b2o$24bo159b2o42b2o11bo15bo20b2o$23b2o158b2o43b4o7b3o15bobo18bo$
185bo43bo2bo5bo19b2o16bobo$133bo40bo54bo2bo5b2o36b2o$93b2o38b3o36b3o
38b2o14b3o$32bo8b2o21bo29bo41bo14bo4bo14bo41bo48b2o$31bobo7bo22b3o27bo
bo38b2o12b3o4b3o12b2o38bobo47bo2bo$30b2obo5bobo25bo27b2o2b2o47bo10bo
47b2o2b2o48bo2bo$32bo6b2o4b2o19b2o31b2o13b2o32b2o8b2o47b2o51b5o$23b2o
20bo15bo51bobo125b2o14b2ob2o$24bo18bobo15b3o51bo59bobo44b2o17b2o16bo$
24bobo16b2o19bo74b3o5b2o10b2o14bo2bo43b2o$25b2o36b2o36bo26b2o9bo2bo4b
2o10b2o14bo2bo$100bo27b2o8bo4bo32b2o$99bo2bo2b2o34bo79b2o47b2o$99bo4bo
117bo47b2o2b2o$99b2o3b3o32b2o78b3o12b2o38bobo$101bo37bobo77bo14bo41bo$
62b2o75bobo93b3o7b2obo27b2o$61bo2bo14b2o12b2o36b2o42b2o36b2o22bo7bob2o
$62b2o15b2o11bobo16b2o19bo42bo19b2o16bobo$63bo28bo18bobo15b3o44b3o15bo
bo18bo$91b2o20bo15bo48bo15bo20b2o$31b2o47b2o25b2o4b2o19b2o36b2o19b2o4b
2o6b3o$27b2o2b2o47bo26bobo25bo36bo25bobo5bo$26bobo38b2o12b3o25bo22b3o
38b3o22bo7bo2bo$26bo41bo14bo16b2o7b2o21bo42bo21b2o8b2o$25b2o27bob2o7b
3o32b2o$54b2obo7bo2$91b2o122b2o$92bo122bo$91bo124bo$91b2o122b2o$151b2o
2b2o$152bo2bo$151bo4bo$151b2o2b2o3$142b2o$111bo21b2o7b2o19b2o8b2o21bo$
109b3o22bo27bo2bo7bo22b3o$108bo25bobo28bo5bobo25bo$108b2o19b2o4b2o25b
3o6b2o4b2o19b2o$114bo15bo20b2o2b2o20bo15bo$112b3o15bobo18bo4bo18bobo
15b3o$111bo19b2o16bobo4bobo16b2o19bo$111b2o36b2o6b2o36b2o$102bobo$102b
obo37bo$103b2o32b3o3b2o$139bo4bo$102bo34b2o2bo2bo$100bo4bo8b2o27bo50b
2o$95b2o4bo2bo9b2o26bo50bo2bo14b2o$95b2o5b3o88bo2bo14b2o$194bobo2$94b
2o47b2o18b2o47b2o$95bo47b2o2b2o10b2o2b2o47bo$92b3o12b2o38bobo8bobo38b
2o12b3o$92bo14bo41bo8bo41bo14bo$108b3o7b2obo27b2o6b2o27bob2o7b3o$110bo
7bob2o64b2obo7bo!
```

P.S., later that day: I had messed up the formulas for P(n) and the other relevant generations. I've fixed that now.
P.S. (3/26/2016): Bill Gosper reminded me that the figure 8 oscillator used in the HWSS synthesis can be replaced by a boat, a reaction found by David Buckingham. So the sawtooth can be reduced slightly:

Code: Select all

```
#C Sawtooth with expansion factor 10/3, based on zdr's c/10 copperhead
#C spaceship.
#C A p112 shotgun produces salvos consisting of a HWSS and two LWSSs.
#C Usually the HWSS is deleted by a glider, but occasionally one is
#C allowed to escape. When it does, it eventually hits the back of
#C the copperhead, which starts a loaf being pulled back toward the
#C shotgun. When the loaf reaches the shotgun, it's deleted, in one
#C of 3 different ways, and another HWSS escapes.
#C Because the e.f. is not an integer, there's no simple formula for the
#C generations when particular things happen. However, a recursive formula
#C exists: Define P(0)=245 and P(n+1) = (10 P(n) - c(P(n) mod 3))/3, where
#C c(0)=735, c(1)=1435, and c(2)=1295. Then a HWSS hits the back of the
#C copperhead in generation P(n). The loaf is destroyed in generation
#C (8*P(n)-d(P(n) mod 3))/3, where d(0)=951, d(1)=1181, and d(2)=1069.
#C The minimum repeating population is 2082. It occurs in generation
#C (8*P(n)-e(P(n) mod 3))/3, provided that P(n) mod 3 is 0 or 1, where
#C e(0)=1512 and e(1)=1736. The successive maximum populations occur
#C around generations P(n) and are approximately 3*P(n)/70.
#C Dean Hickerson, 3/8/2016. Reduced with Bill Gosper's help, 3/26/2016.
x = 280, y = 250, rule = B3/S23
206bo7bob2o$204b3o7b2obo27b2o$188bo14bo41bo$188b3o12b2o38bobo$191bo47b
2o2b2o$190b2o47b2o$210bo$211bo$102bo7bob2o77b2o18bo$100b3o7b2obo27b2o
48b2o17b2o$84bo14bo41bo$84b3o12b2o38bobo65b3o$87bo47b2o2b2o$86b2o47b2o
2$98bob3o$87b2o10bo2bo104b2o36b2o$87b2o13bo3b2o99bo19b2o16bobo$100b2o
4b2o100b3o15bobo18bo$98b2o110bo15bo20b2o$204b2o19b2o4b2o4bo$204bo25bob
o4bo$205b3o22bo6bo$133bo3bo69bo21b2o8bo2bo$103b2o28bo2bo4b2o96b3o$31b
2obo7bo60bo19b2o9bo2bo3bobo$2b2o27bob2o7b3o59b3o15bobo9bo2bo5bo$3bo41b
o14bo45bo15bo12b3o5b2o102b2o$3bobo38b2o12b3o39b2o19b2o4b2o118bo$4b2o2b
2o47bo42bo25bobo119bo$8b2o47b2o42b3o22bo120b2o$103bo21b2o7b2o$134b2o
47bo$40bo15b2o125bobo$40b2o14b2o125b2o$40bobo100b2o$143bo$144bo49b3o$
143b2o48bo2bo8b2o21bo$83b2o113bo6bo22b3o$83bo114bo4bobo25bo$2b2o36b2o
42bo113bo4b2o4b2o19b2o$bobo16b2o19bo41b2o102b2o20bo15bo$bo18bobo15b3o
147bo18bobo15b3o$2o8bo11bo15bo149bobo16b2o19bo$9bo6b2o4b2o19b2o47b2o
95b2o36b2o$8b3o5bobo25bo47b2o7b2o21bo$18bo22b3o57bo22b3o$18b2o21bo57bo
bo25bo$99b2o4b2o19b2o$83b2o5b3o12bo15bo104b3o$84bo5bo2bo9bobo15b3o$2o
82bobo3bo2bo9b2o19bo99b2o17b2o$bo83b2o4bo2bo28b2o99bo18b2o$o61bo27bo3b
o129bo$2o61bo161bo$61b3o131b2o47b2o$191b2o2b2o47bo$128b2o60bobo38b2o
12b3o$120b2o4b2o62bo41bo14bo$120b2o3bo13b2o14bo33b2o27bob2o7b3o$125bo
2bo10b2o14bobo60b2obo7bo$125b3obo25b2o$20bo21b2o$18b3o22bo47b2o47b2o
84bo7bob2o$17bo25bobo5b3o33b2o2b2o47bo83b3o7b2obo27b2o$17b2o19b2o4b2o
6bo33bobo20bobo15b2o12b3o64bo14bo17b2o22bo$23bo15bo11bo8b2o24bo23b2o
16bo14bo64b3o12b2o15b2o21bobo$21b3o15bobo18bo24b2o23bo14b3o83bo30bo16b
2o2b2o$20bo19b2o16bobo64bo84b2o8b2o37b2o$20b2o36b2o164bo$221bo3bo$211b
2o12bo$211b2o7bobob2o4b2o$222b2o6b2o2$19bobo$4b2o14b2o237bo3bo$4b2o15b
o239bobo$258b2o$227b2o29bo6b2o$3b2o47b2o173bo19b2o10bo2bo2bobo$4bo47b
2o2b2o32bo137b3o15bobo10b3o5bo$b3o12b2o38bobo32bo138bo15bo20b2o$bo14bo
41bo30b3o132b2o19b2o4b2o$17b3o7b2obo27b2o164bo25bobo$19bo7bob2o194b3o
22bo$227bo21b2o7b2o$46b2obo7bo69bo130b2o$17b2o27bob2o7b3o67bobo$18bo
41bo14bo51b2o$18bobo38b2o12b3o191b2o$19b2o2b2o47bo194bo$23b2o47b2o194b
o$53bo213b2o$52bo$52bo18b2o$52b2o17b2o192bo$265b2o$54b3o207bobo2$216b
2o$216b2o7b2o21bo$225bo22b3o$17b2o36b2o166bobo25bo$16bobo16b2o19bo166b
2o4b2o19b2o$16bo18bobo15b3o151b2o20bo15bo$15b2o20bo15bo154bo5b3o10bobo
15b3o$26bo4b2o4b2o19b2o148bobo2bo2bo10b2o19bo$26bo4bobo25bo149b2o6bo
29b2o$26bo6bo22b3o157b2o$21bo2bo8b2o21bo155bobo$22b3o187bo3bo3$15b2o
227b2o6b2o$16bo82bo144b2o4b2obobo7b2o$15bo83bobo148bo12b2o$15b2o82b2o
149bo3bo$251bo$80bo134b2o37b2o8b2o$78bobo130b2o2b2o16bo30bo$79b2o129bo
bo21b2o15b2o12b3o$210bo22b2o17bo14bo$209b2o27bob2o7b3o$67b3o168b2obo7b
o$35bo21b2o8bo2bo$33b3o22bo6bo$32bo25bobo4bo$32b2o19b2o4b2o4bo$38bo15b
o20b2o$36b3o15bobo18bo$35bo19b2o16bobo95bo$35b2o36b2o3b4o89b2ob3o$77bo
4bob2o80b2o7b3o$77bo4bo3bo81bo4b2o3bo$80b2o4bo81bo4b2o3bo$77b2o4bo2bo
79b2o7b3o$35b3o38bo2bo3bo2bo3b2o79b2ob3o$76bo2bo4b2o3bobo79bo$19b2o17b
2o36bo4b2o7bo$19b2o18bo36bo3bo4bo$39bo37b2obo4bo$38bo42b4o164b2obo7bo$
18b2o47b2o151b2o27bob2o7b3o$19bo47b2o2b2o148bo41bo14bo$16b3o12b2o38bob
o147bobo38b2o12b3o$16bo14bo41bo148b2o2b2o47bo$32b3o7b2obo27b2o151b2o
47b2o$34bo7bob2o2$274b2o$42bo7bob2o184bo16b2o17b2o$40b3o7b2obo27b2o
153b2ob2o14b2o$24bo14bo41bo151b5o$24b3o12b2o38bobo151bo2bo$27bo47b2o2b
2o152bo2bo$26b2o47b2o157b2o$266b3o$44bo175b2o36b2o5bo2bo$27b2o15b2o
173bobo16b2o19bo5bo2bo$27b2o14bo2bo172bo18bobo15b3o7b4o$44b2o172b2o20b
o15bo11b2o$234b2o4b2o19b2o7bo$234bobo25bo6bo$236bo22b3o7bo$227b2o7b2o
21bo$227b2o$43b2o36b2o$43bo19b2o16bobo$44b3o15bobo18bo$46bo15bo20b2o$
40b2o19b2o4b2o6bo$40bo25bobo5bob2o$41b3o22bo7bobo142b2o57b2o$43bo21b2o
8bo143b2o58bo$278bo$278b2o3$269b2o$85b2o151bo21b2o7b2o$86b2o140bo7b3o
22bo$23b2o60bo142bo6bo25bobo$23bo203bo7b2o19b2o4b2o$24bo159b2o42b2o11b
o15bo20b2o$23b2o158b2o43b4o7b3o15bobo18bo$185bo43bo2bo5bo19b2o16bobo$
133bo40bo54bo2bo5b2o36b2o$93b2o38b3o36b3o38b2o14b3o$32bo8b2o21bo29bo
41bo14bo4bo14bo41bo48b2o$31bobo7bo22b3o27bobo38b2o12b3o4b3o12b2o38bobo
47bo2bo$30b2obo5bobo25bo27b2o2b2o47bo10bo47b2o2b2o48bo2bo$32bo6b2o4b2o
19b2o31b2o13b2o32b2o8b2o47b2o51b5o$23b2o20bo15bo51bobo125b2o14b2ob2o$
24bo18bobo15b3o51bo59bobo44b2o17b2o16bo$24bobo16b2o19bo74b3o5b2o10b2o
14bo2bo43b2o$25b2o36b2o36bo26b2o9bo2bo4b2o10b2o14bo2bo$100bo27b2o8bo4b
o32b2o$99bo2bo2b2o34bo79b2o47b2o$99bo4bo117bo47b2o2b2o$99b2o3b3o32b2o
78b3o12b2o38bobo$101bo37bobo77bo14bo41bo$62b2o75bobo93b3o7b2obo27b2o$
61bo2bo14b2o12b2o36b2o42b2o36b2o22bo7bob2o$62b2o15b2o11bobo16b2o19bo
42bo19b2o16bobo$63bo28bo18bobo15b3o44b3o15bobo18bo$91b2o20bo15bo48bo
15bo20b2o$31b2o47b2o25b2o4b2o19b2o36b2o19b2o4b2o6b3o$27b2o2b2o47bo26bo
bo25bo36bo25bobo5bo$26bobo38b2o12b3o25bo22b3o38b3o22bo7bo2bo$26bo41bo
14bo16b2o7b2o21bo42bo21b2o8b2o$25b2o27bob2o7b3o32b2o$54b2obo7bo2$91b2o
122b2o$92bo122bo$91bo124bo$91b2o122b2o$151b2o2b2o$152bo2bo$151bo4bo$
151b2o2b2o3$142b2o$111bo21b2o7b2o19b2o8b2o21bo$109b3o22bo27bo2bo7bo22b
3o$108bo25bobo28bo5bobo25bo$108b2o19b2o4b2o25b3o6b2o4b2o19b2o$114bo15b
o20b2o2b2o20bo15bo$112b3o15bobo18bo4bo18bobo15b3o$111bo19b2o16bobo4bob
o16b2o19bo$111b2o36b2o6b2o36b2o$102bobo$102bobo37bo$103b2o32b3o3b2o$
139bo4bo$102bo34b2o2bo2bo$100bo4bo8b2o27bo50b2o$95b2o4bo2bo9b2o26bo50b
o2bo14b2o$95b2o5b3o88bo2bo14b2o$194bobo2$94b2o47b2o18b2o47b2o$95bo47b
2o2b2o10b2o2b2o47bo$92b3o12b2o38bobo8bobo38b2o12b3o$92bo14bo41bo8bo41b
o14bo$108b3o7b2obo27b2o6b2o27bob2o7b3o$110bo7bob2o64b2obo7bo!
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