itch.io

Some side thoughts:

1. I have over 200 books! It's getting tedious to scrap them all, I want a way to bulk scrap them. Maybe make the books stackable?

By the way, I think some inventory sorting function could be useful, so things are in standard order and not change when I use up something and gain more later.

2. With about 100 maps explored, the selection screen is getting cluttered. Now I really want a way to delete old locations.

3. When exploring in the mines, I have an idea: When muffled, the character can use an action to make noise (like knocking the ground), so the centipedes are attracted together while I quietly move to another position, then I can use a single potion to kill a group of them.

4. About Verdant sandals:

This equipment is useless outside of setting forest fires. And even when I'm using it to prepare a forest fire, it's still inconvenient to use.

I think it should inherit the nettle protection from tall boots, as provided by verdant blessing.

If it needs to be buffed, then either give it better utility for preparing the fire, or give it some other utility.

For preparing the fire, it could either be buffed to grow tall grass in a larger range (3x3), or be able to overwrite non-flammable plants like spearmints and foxgloves.

For other utility, it could give hp recovery. Maybe heal 1 hp for every tile of tall grass planted. If that's too much, you can make it heal with a probability.

5. Roses hurt!

Currently rose deal 1 damage when passing through, and no equipment can block this damage. The pathfinder also doesn't try to avoid it. I think this needs to be fixed, either make tall boots block that damage, or make the pathfinder avoid it.

Now I've (almost) completed the game, it's time for me to write a review.

(spoiler warning)

TLDR: Overall, I think the update is great, and still needs polishing. Some balace still need tuning.

1. Centipede and Ghoul droppings are too weak, and don't match their strength.

Centipede drop only gives 1 poison when distilled. For comparison, a toad drops a slime gland, which also gives 1 poison when distilled.

I suggest buff the centipede drop to 2~3 poison.

Ghoul drop gives 0.4 sanguine when distilled. For comparison, a Leech pouch gives 0.5 sanguine when ground in a mortar.

The ghouls are rather frightening, I suggest change them to give sinister instead. Compared to the abundant spider silk (0.25) and fiend powderbox (0.4), and another frightening monster Chimaera (1), I suggest they should give 0.6~1 sinister.

2. Efreet and Centipede show the design intention that I should avoid them, but in practice, they have high speed, and there is often not enough space to sneak by, so I have to kill them. This way I'm no longer outwitting my enemies, instead I'm brute forcing them with my vast potion supplies, and this feels a little wrong.

I suggest add 'dashing' equipments, which can help with sneaking past these enemies. Dashing could work like "the stardust", speed up movement but cost hp every move, and maybe accumulate fatigue so the player can't dash indefinitely.

3. Exploration of T3 fey lake is almost identical to T2 forest, aside from the fairy and the moat.

I hope the forest can have some new threat than the T2 forest, rather than having the riddle be the only difficult part. Maybe some normal creatures like bears? Or some mystical fey creature?

4. Where is the sulfur? So far I have only seen sulfur in a variant of T1 farm, and nowhere else. I hope there are more sulfur later.

5. Flame spire have some rooms with wooden barrels, that start to burn very soon after I enter the map. It's as if they are in chronostasis before I entered the map, which breaks the immersion. I think they should either be burnt at start, or simply not flammable, or be locked well and have no efreet inside.

This save has 2.5 years, and have hoarded a huge amount of items. I have 2k gold, 8k fuel, 1k metal, 12 complete map and 6 unexplored T3 zones. I also have enough materials to make 20 elixir of youth.

Although there is no time contraint, I still try to optimize in-game time usage, and use this to estimate potion costs.

The T3 maps: 

Sorting in difficulty, I think it's Spire = Mine = Grave >> Lake. The lake riddle is not taken into account, because the tree trunks give a huge amount of reward without needing to complete the riddle.

I have completed 3 Spires, 5 Mines, 5 Graves, 4 Lakes (but failed all the riddles).

The forest part of spire:

The forest is all burned, and I need to avoid the Efreets. The good part is, the forest is all burned, so the vision is clear, and I don't need to worry about being ambushed around a corner.  There is usually enough space to maneuver around them, and I can destroy burned trees to create paths when in need.

It's relatively easy to reach the spire without needing potions. However, the forest have some good materials, and to collect those, I need to kill some Efreets. The rule of thumb is, use a frost potion whenever it could kill 2 Efreets or more.

The magmoss is useful, it's a good source of telluric. 

The scorch leaves is less useful, it can be refined into either lawless or volatile, but neither are important.

The smoulderfly is simply not worth gathering. It requires hushing sands to gather, but only gives 1.5 volatile, which is already abundant. I suggest buffing it to 2~3 volatile, and maybe make it take sorting instead of grinding.

Inside the spire:

My biggest problem with it, is the rooms with wooden barrels. They start to burn soon after I enter the floor. When I reach the rooms, I usually see them burn in front of me.

In my opinion, they should either have already burned up before I come to the spire, or be entirely immune to Efreets. Maybe the barrels and chests should be made of metal, or some magical crystal that can be collected for later use.

In the spire, there is usually no longer enough space to sneak around the Efreets. I simply use frost potion to kill all of them. The rule of thumb is, use a frost potion whenever it could kill 2 Efreets or more, and also if a single Efreet is blocking the path.

On average, I use about 20 frost potions each spire.

I also tried using Corpselight potion against Efreets, because the potion description says "freeze all nearby creatures", but it seems to have no effect on them.

Pirate graves:

These are mostly fine imo. 

The Ghouls are designed as need to be killed. Each cost 3 fireballs, or 1 black fire + 1 fireball.

The ghoul drops are underwhelming compared to their strength.

On the ground, it's possible to cheese them with crystal potion. I can use a crystal potion on each ghoul and then push them into deep water.

Sometimes, it's possible to crystal a ghoul and put it in a side room. If it has lost track of the player, it will stay dormant and not try to break open the doors. Not having to kill it can save some potions.

The 3 sarcophagus can be solved with the whisper candle, but I always drink a phoenix potion before opening it, just in case.

Mines:

The entrance has renewable sources of glacial water. However, I find it not worth to go there again just for that water.

The mines require having hushing sands active almost all the time. I also use restlessness to make hushing sands last longer, and use food instead of calming infusion to recover health.

The centipedes are deadly. They have 130% speed, and deal 4~6 damage on bite. After the bite, there is also a 50% slowdown, which makes it hard to either continue sneaking or retreat. 

There is usually not enough space to sneak around them. With muffled footsteps, they act similar to Efreets, but are somewhat harder to kill.

Sometimes there are very tight corners which blocks most of vision. A centipede can be very deadly in this case.

It's possible to leave some of the centipedes alone, but they usually guard some treasure and valuable ore behind, so I still kill them.

Killing each takes 1 fireball + 1 embers. A single black fire potion can also do the job, and becomes the best option when it can affect 2 or more targets.

The bottom floor has lots of pits. It's possible to use crystal potion + push to kill centipedes. The wyvern is no more difficult than the fort one, so it's just a piece of cake.

When exploring the bottom floor of water variant, I noticed a problem: When I cling to a wall with gecko potion, deep water have no movement penalty, but shallow water still have movement penalty. I think this needs to be fixed. 

Fey lake:

The lake can be passed through costing 2~3 floating potions. There's nothing interesting about the lake itself.

The forest is almost identical to the T2 forest. There are no additional threats. There are additional resources, the mandrake and the resources in tree trunks. The fairy dust and strange drinks are all useful materials, and if I fully explore the map, it can give far more than 4 map fragments.

The fairy riddles are very difficult for me. So far I have tried 4 times, and no success. Some riddles are clear enough that I'm confident of the answer, but some are vague, and I need to guess and end up failing. Maybe I need to ask ChatGPT for help?

I'm not asking for a nerf to the riddle difficulty, I just hope there's nothing important locked behind them in

The hotbar is a huge QoL improvement, now I can't live without it. Unfortunately it only has 10 slots, which is not really enough for late game builds.

It partially solved the frequent equipment switching problem, by making the switching easier.

There is a cheese with repeatedly turning lights on and off to trick bats. It already existed in old versions, but the hotbar makes it actually viable.

The ability to unequip potion in also very helpful, especially when combined with the hotbar. It has prevented misclick many times.

New Items (and changed items):

1. Black fire:

This update buffed its damage, and added 2 types of enemies that can be affected by it: Ghouls and Centipedes. Black fires now deal about 1.8x damage of fireballs at 2.4x cost, which makes it a direct upgrade.

While this sounds like less cost-effectiveness, it can make good use of oppertunities when a single potion can affect multiple targets. The spread is still hard to use, but it's good enough without the spread. Overall I think it's well-balanced.

2. Unruly flame:

Its damage is too unstable and unreliable for main dps. I had a try at it, but it can't even reliably one-shot ghouls. At this price, it's not very useful.

3. Rime potion:

It's now in a strange position. Currently it's used only in the spire, but it's unlocked very early, and the spire is a late game T3 map. So there is a very long time, I have unlocked rime potion but have no need for it.

4. Chaos potion:

For the drinking and applying part:

When I want to apply a buff on myself, in most cases, only 1~2 of the buffs are needed. The chances of rolling the needed buff is low. For drinking, it's a little better, because it can give healing, but this usually results in overflow, so it's not useful either.

For the throwing part:

It's possible to use this as a damage potion. While there are many different thrown potions, a good number of them deal damage in some way, so there's a good chance to deal some damage.

However this has some problems. The first problem is the rift potion. You can accidentally exchange your position with the target. So it's only useful when fight a single enemy. Otherwise you may end up in the middle of a group of enemies, or dangerously close to a strong enemy. The second problem is immunities. Some enemies can be immune to some of the potions, reducing the chance of dealing damage.

I've tried using chaos potions to deal with single centipedes in the mines, but the result is far from ideal. It usually takes more than 6 to kill each.

The cost of the potion is not trivial. Overall it's still not useful.

5. Poison gloves:

The passive effect is very useful. It allows curing rot effect, which is useful for the ghouls in graves.

The melee effect is somewhat useful. I can now kill bats in 1 hit, which needs less turning lights on and off.

The poison cloud buff is useless. All major threats in T2 and T3 maps are immune to poison.

6. Restlessness potion:

This potion multiplies player speed by 1.15x. This is not useful for sneaking past Centipedes and Efreets, because the player has no way to tell when the extra turn will occur.

The main use of it, is to pair with hushing sands and gecko potion, so these potions last longer.

The loss of calming infusion is actually not a big problem. Calming infusion often results in healing overflow, so it usually heals less than the theoretical 60 hp. Food from richardo is cheap enough to be a good alternative.

7. Dream catcher: This is rather helpful, as it can generate lots of high tier materials. Holy, Sinister and Semipiternal are all very useful.

8. Phoenix potion: This seems to be a specialized potion for solving the pirate grave. I usually keep a few in inventory, but not use them often.

9. Whispering candle:  This item reveals some interesting lores. I have tried listen to them in each type of map. They are mostly useful in pirate graves, to steal the secret from the pirate. In other places it's mostly just cool.

10. Potion of blood transfiguration: This potion is more expensive than fireball, and fireball can do its job nicely.

11. "The Stardust": The Fey lake area gives tons of fey essence (hundreds), so I can brew lots of them to sell. Still have nowhere to spend that much money though.

12. Shadowbane potion: It's throw effect is useful at clearing densely packed shadow clouds. However, it's also ok to just rush through the cloud, it only costs 3 hp.

The drink effect lasts too short to be useful. I think it should be at least 40 turns to be a worthy alternative.

13. Kingsblood: Works as intended, but not useful outside of unlocking their locks.

Kingsblood can be used to fool the living armor of the T2 mansion. However this doesn't work in practice, as 15 turns is not enough to loot the main treasure. I can use 2 potions, but it's no longer cost-effective this way.

14. Floating: Works as intended, but no use outside of the fey lake. Duration is too short for other use cases.

I haven't used either potion of recollection or discord. I think discord could be useful when it's affecting 2 ghouls or centipedes, but haven't tried yet.

Richardo T2 shop adds something useful in late game. The important goods are: Metal sun symbol, ivory dice, ink, black feather. Bloodstone and black quartz can be useful, but I found them not cost-effective.

It's getting tedious to check the shop every in-game day. I hope the T2 shop can have bigger capacity and restock slower.

I'm playing on steam. When playing with the whispering candle, I noticed the whispers are missing for the T2 forest.

I'm playing the new version (paid on steam) and started a new save.

It seems there is a change in the book loottable, which makes it harder to deliver potions to the travelling merchant in time. I also accumulated ~12 academic knowledge in mid-game, before being able to use them in the new techs.

The new t3 maps are really difficult. I have already died twice in the flame spire, mostly because I went there too early. Luckily I picked the infinite life option. I think they will become easier when I get used to them, but will still cost a lot of potions to explore.

I encountered Raemus heir. In the conversation, he forcefully took 5 piece of gold from me. (he asked for 15 but I only had 5 at that time) 

I wonder whether this is a scam, or is there some more story behind him?

5 pieces of gold is not a big deal, but I'm angry that someone can rob money from me and then show up as if it's completely justified. If this is really a scam, then I want him DEAD.

I wonder whether we can get some form of curse in the later stages of the game, to deal with this, or hire assassins to do this. Richardo is already doing those shady things (the 4 potions are probably used for them), and our alchemist should be able to afford such a cost in mid game.

Comment about the ingame government balance:

Socialism is OP in this game. It turns this game into a critter breeding game. Always aim for it first, the other 2 types of economy are completely useless. 

Suppose you have 2 parent cards, A and B, then use them to breed C. Put C in that representative to boost A and B, then discard C after an empty draw. Rinse and repeat. This is exponential growth until it hits the 999 hardcap. It should be able to reach all 999 stats with around 150~200 turns.

For political system, totalitarianism is the most useful. With it, you can increase influence of a card from 1 to 2 in academia, then reduce to 0. This allows you to reduce influence production of everything to 0, greatly easing management.

None of the 3 tribal systems are important. Caste system could be useful if you have "Workplace shift" prestige upgrade.

Buy it first, and make money from renewable sources (food and fabric), you'll be able to make your own weapons later on, I think you need to push through 10 floors to get it

Reached current endgame at 3000 years. Previous endgame was 1700 years. This update is certainly a lot faster.

Played a bit more, realized the relics do nothing other than unlocking breakthrough. oops.

The Raid relic rewards have no descriptions. I just get the reward and know nothing about it. Maybe add some interface somewhere, to view all known relics? I usually enjoy looking at the formulas and trying to predict how the upgrades interact in my head.

"Diamond infused with jade" and "Elder training scroll" seem to have their icons wrong

How is the development going? The release date on steam is close, will it be postponed again?

I don't like the storage cap. It adds little to the game, but creates a lot of frustruation. The worst part is how it interacts with base modules: Normally, deconstructing a building refunds 100% of cost. But when I hit the storage cap, boom, all of that module is lost. No warnings in advance.

What is gold used for?

Oh no, steam release is postponed again...

I think 4) is the lack of QoL. In my playthrough, the game become quite repetitive after 5 holdings. I reached 10 holdings and unlocked multi-holding, but it didn't help towards grinding projects.

I think this game needs some milestone to keep more finance on holding reset.

IMO keeping the HR/IT buymax and keep will be enough. The holding reset should still reset everything in these 2 layers, but the milestones should be kept. Keeping everything in these 2 layers is too OP though.

I played the game for a while (1d ingame), and reached 14k dps. I had quite some fun, and I also noticed some problems.

1. The storage limit mechanism doesn't interact with the rest of the game in a meaningful way. 

I suggest simply removing it. Even without storage limits, there are still many places to spend the money.

2. The pacing is way too slow. It's so slow, that it breaks a major part of game balance. 

At my current stage, it takes multiple hours to get anywhere, and it seems things will slow down even more as I progress. 

3. Manual mining is fun, but the game strongly discourages manual mining. This is mainly caused by the slow pacing. With things taking multiple hours to get anywhere, manual mining is simply irrelevant.

The game has quite some original designs, but they all depend on manual mining.

4. The weapons mechanism is interesting, but it's also affected by the balancing problem.

From what I see in the game, manual mining overrides fire rate by a constant (about 3/s), and by fast tapping space it can fire even faster. This makes slow firing weapons suitable for manual mining, and machine guns actually fire slower with manual.

For T2, Shotgun is good for manual mining, and still deals some dps for auto. Machine gun is optimal for auto but deals little damage in manual. 

Puncher is even more powerful on the paper. With green ammo, it has higher auto damage than machine gun. However, it costs a lot of power, which severely limits its use. Also, its AOE is hard to utilize in manual mining. Imagine a puncher bombing hard at a dense ore block, and all blocks below it is already empty. It's also more expensive than other T2 weapons, which means it's pretty impractical.

T3 introduces another choice: Charged shot. It has decent dmg for both auto and manual, and is the cheapest T3 weapon, but requires a pool of excess power to work properly. The same pool of excess power could be reused by multiple Chargers, so it's really powerful.

I really like the weapons balance of this game. However the slow pacing makes manual mining irrelevant, and most of this weapons balance goes out of the window.

5. Coolant seems useless. They are not worth their equipment slot.

T1 coolant only reduces heat by 10%, and I could simply just add another plating. T2 is 20% and faces the same problem. Maybe higher tier coolant could be useful.

I have 2 suggestions for this:

 a. Make all tiers of coolant more powerful, and nerf plating to compensate.

 b. Add a power cost to them. This will make coolant upgrade a more impactful choice.

6. Charged shot description says it depends on excess power in the rig.

However, the effect of excess power isn't continous. Increasing power is only useful if it crosses a certain threshold. These thresholds are currently hidden. I suggest the game should show these thresholds.

It took me a while to find out how this works. I was confused at first and only figured this out after jumping into a chasm and crashed to death.

You don't return to your base safely. Instead, you return your base when you die, either by running out of hp or fuel. This is the normal way to return, and has no negative consequences.

After dying and returning to base, you can craft there.

No jumping, but there is ladder for climbing. They cost resources so you shouldn't use them too much.

You need to beat the #100 enemy of z10 to start getting that new resource.

You should push to higher zones (~z14), and come back when your attack is about 10x of first enemy hp of z10.

I think the code is working correctly but the description is wrong. It should be halving impurity penalty, not impurity.

Edit:typo

That's making things even worse. My post complained about things limiting idle play, and you said about things require idle play. Both type of games exist, but when these two things are in the same game, it makes the game really torn. If the game is designed to be played neither idle or active, then how do I play the game?

Is the game really designed to be played idle?

The game description says yes, but the game mechanisms say no.

I'm at level 132, and from what I see, this game strongly penalizes long idle periods by having storage limits. And even worse, in the horde, I have to constantly buy upgrade to be able to make any progress. The farm also requires very frequent planting.

This problem is especially significant in the horde. After ~Z36, I basically run out of bone upgrades to buy. The infinite upgrades alone are not enough to keep me going. So I can't get any further, and the only option is to prestige. This means a long run isn't much better than a short run.

In a short run, I can reach Z38 in 40 minutes, earning ~120 soul. If I keep pushing, I can get to Z41 in half a day, but the reward is only ~180 soul. This is nowhere near enough to make long runs viable, and the new unlocked upgrade is also unaffordable.

From what I see, the only way out of this game design disaster is automation. However the only automation are mining autoprogress and auto horde boss. Not even autobuy upgrades. And many mechanisms are designed in a way that is very difficult to implement automation.

I'm afraid so. Most of the sub features are not useful when unlocked, and takes a huge amount of time investment to actually start having any benefit, and even when they start having benefits the bonus are low. This really isn't good game design.

I suggest the game should show the price increment for each upgrade. This allows for better planning of prestige upgrades and gem upgrades.

For the gem mechanism, I think it should be immediately useful when unlocked. The game can give player some gems on unlocking it, and also give gems as achievement rewards. These should only give red and purple (and maybe green) but never blue gems.

I think 150 red gems at start should be enough, it allows the player to choose one of the upgrades.

How large is the spawning area of the spawner? It's not said in the game but I really need it for a good spawner automation.

Got all upgrades. Only caught 35k pixels, nowhere near 5m. Lol

I downloaded it manually, and played for a while. I encountered another bug.

After buying 3 funny upgrades, the Thing button is broken. It says 'you get <null> things per click', and if I click it my things are broken. I had to reload because of this.

The 1000 cyyan upgrade is broken because it is based on manually clicking. 

I still managed to get to the end, it seems gain from clicking is irrelevant anyways.

The title says it all.

I think the cause of the problem is, the windows version file is not marked for windows.

Yes, I pressed that reset button after I found out I wasted too much fuel with the combustion engine.

If you decide to keep those obsolete techs in the tech tree, I think you can add the ability to hide certain techs. This prevents too many obsolete techs from cluttering. 

If the combustion engine is kept, I think it needs a redesign: Since the fuel efficiency is all what matters, you can increase its base efficiency a lot, but make it not scale proportionally to number of engines. This makes it useful at start, and fusion is still a big improvement.

Alternatively, the real time research speed could be based on proper velocity, so increasing velocity early is still helpful, allowing an early engine to exist.

Another note: When fusion is unlocked, it's relatively easy to research the efficiency curve to high levels. This basically makes that efficiency irrelevant because it's near 100% all the time. This is even more true for antimatter engines. So I think the whole efficiency mechanism should be removed for the fusion and antimatter, and leave it exclusively to combustion engines. 

The tutorial shows an overwhelming amount of information at start, and I have no way to look it up later on. Please add something in the options so I can look it up when needed.

I have reached current endgame, with 600k lorentz factor and lvl 93 antimatter engine mass & auto research. I have ~600k antimatter engines.

I have 103684m2 of fuel net, and L41 antimatter engine consumption. Fuel gain scales really fast with speed, and with any more net I will start gaining fuel faster than I can use them. I will eventually get too much fuel that weigh my ship down.

Looks like fuel net has 'air resistance', so the terminal speed depends on how effcient you use the fuel. Strangely, the fuel effciency is boosted by the number of engines. 

3 suggestions:

1. Combustion engines are useless. I would better skip them, because they can't gain any meaningful speed to get lorentz factor significantly above 1. The game only starts when I get to fusion.

From lore perspective, combustion engines are useless if you want to get anywhere near c.

I think you should directly remove combustion engine tech, and start the player with fusion, and maybe add 'exotic' after antimatter.

2. I think fuel should have a limit, so fuel doesn't weigh the ship down.

3. I think it would be interesting to add some way to gain more building materials.

I have  a similar problem: one of my worker has negatice energy and the game stops running. I think I have to hard reset.

After I started charging the command room, it drains all energy on the grid, preventing me from increasing my power generation any further via the propulsion room. I didn't have a previous save. I think this needs to be fixed.

I think the manual smashing missions need to have their goals nerfed. It's really tedious to manually smash 777 big mugs, and if my save gets reset, I certainly won't want to do this again.

Stone generates tech points based on sum of production around it, so you need to put it next to your production units. I think the tutorial isn't clear enough.

just honeycomb glacier and forest. I have 4 forest and 8 glacier, and I'm getting more than 7k tech/s, which is 1e5 research in about 300 secs.

Nice! I always wanted to buy this, but I didn't have the payment supported by itch. I'll definitely buy it when it comes on steam.

I think you can make the player "softlocked" by missing a certain skill, but at the same time, enable inverting a craft, so the player can get out of the softlock.

Inverting a craft should cost nothing, but it still takes time to craft it back.

Very interesting game. I've reached my first prestige, and it seems to become much more grindy from this point.

I have encountered one problem:

Prestige resets offline time.  I have played many games with offline time mechanisms, and none of them reset offline time on a prestige.

I think offline time is a savefile layer thing, so only hard reset should reset it.

Interesting concept, but I don't see the relation between collatz conjecture and incremental mechanisms.

Currently, the only incremental mechanism in the game that makes use of the conjecture, is how many steps each number will take to go back to 1. This determines how many producer ticks can be made per verifier tick. This is just 'idle clockwork', not 'collatz clockwork'.

I have some ideas for improvement:

1. Concept:

This is just a game, and players' computers are nowhere near enough to find nontrivial results to collatz conjecture. 

Even if a number diverges, the player's computer has no way to determine it's actually a counterexample. This process can't prove the conjecture either.

Limiting the money gain by the player's computer power will also severely limit the incremental growth of the game, and therefore limiting game design.

So you can forget about proving the lower bound, and instead make use of the sequence produced from the numbers. This will allow for more interesting incremental mechanisms, and stays close to the conjecture.

2. 'Level' skips:

Instead of trying to check every number to 1, you can instead skip some numbers that will trivially reduce to some previously checked numbers, like even numbers.

3. Sequence based mechanisms:

It's very simple to design, like 'multiply money gained based on current sequence length', or even 'multiply money by log of highest number reached in the current sequence'. These are much more interesting and also more connected to the conjecture.

These upgrades will benefit from finding a longer sequence that oscillates wildly before eventually converges (or maybe doesn't converge within the available steps). The level skip could completely discard the current verification scheme, and instead focus on searching numbers that leads to such interesting sequences.