General Guide by kaennish
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Table of Contents
1. Introduction2. Gold Efficient Pure Damage
3. Gold Efficient Overall DPS
4. Flat Damage (Bursty)
5. Overall BEST DPS
6. Item Scores (Based on Stat Weights)
7. Stat Weights (Based on Maths)
8. The Maths (Long and Boring)
9. Summary
Introduction
// This is more of a collection of theorycrafted builds than it is a guide. Expect to learn the BEST builds for the champion with math to prove it. Don't expect to learn how to play the champion. Eventually I will add in a guide, probably in video form, explaining how the champion is played with a few tips. Until then enjoy knowing the 100% best builds for the champion proven by maths. // SPECIAL THANKS: Credits to jhoijhoi for the template, which you can find here. 
Gold Efficient Pure Damage Build
// Increased movespeed, and attack speed. Makes it easier to dodge things in lane and also increases the damage of your ultimate, The Culling, as well as your Auto Attack DPS. These are the only boots that will actually give you more flat damage, so its the clearcut choice for this build. 
// This can be a bit risky to get early game because it doesn't give you any damage, but with it you can spam abilities to your hearts content and it upgrades later on into the almighty Muramana which is the best DMG per gold item you can get. So get it semiearly and once it gets full stacks upgrade it. You can also upgrade it earlier if you want to stack it faster, but its not actually worth the gold until full stacks. And this build is all about being worth the gold. 
// Massive spike in damage with this item, if you have enough gold on your first or second time back to base go ahead and get the B. F. Sword. This is the most cost effective item early game, so rush it if you can. 
// This item's stats aren't actually all that great, but because its so cheap at 2300 gold its definately worth it once mid game comes around and people have a decent amount of armor. 
// The only items more cost effective than The Bloodthirster we already have, so the only thing left to do is get another. Another alternative is a Blade of the Ruined King. Which technically isn't as cost effective as this item, but the activate is very helpful so in my opinion its still a good buy. 
For a 6th item go for something with survavability. I can't really tell you which item to buy since tanky items are all situational. If you're having trouble against AD a Randuin's Omen is always a good buy. If CC/magic damage is the problem a Mercurial Scimitar is an amazing item. If its a mixture between magic AND physical you could go for a Guardian Angel or even a Warmog's Armor if you really wanted to.
Gold Efficient Overall DPS Build
// Because Berserker's Greaves cost 100 gold less than Ionian Boots of Lucidity they actually give more DPS Per Gold, so they are the best boots for this build. 
// This can be a bit risky to get early game because it doesn't give you any damage, but with it you can spam abilities to your hearts content and it upgrades later on into the almighty Muramana which is the second best DPS per gold item you can get. So get it semiearly and once it gets full stacks upgrade it. You can also upgrade it earlier if you want to stack it faster, but its not actually worth the gold until full stacks. And this build is all about being worth the gold. 
// Early game this item will give you the most DPS for your gold, so rush it as soon as you can. If you can buy a B. F. Sword on your first or second time back to base get it over the Berserker's Greaves or Tear of the Goddess. Also the early sustain is a huge help. 
// This item is the second best DPS Per Gold item after Blade of the Ruined King. Buy it once your Tear of the Goddess is stacked fully or very close to being stacked fully. The toggle is amazing on this ability so anytime you can spare the mana turn it on. 
// Only 2 items can trump The Bloodthirster for DPS Per Gold and you already have them, so at this point just buy another! Also this will be your third lifesteal item so your sustain will be insane. 
For a 6th item go for something with survavability. I can't really tell you which item to buy since tanky items are all situational. If you're having trouble against AD a Randuin's Omen is always a good buy. If CC/magic damage is the problem a Mercurial Scimitar is an amazing item. If its a mixture between magic AND physical you could go for a Guardian Angel or even a Warmog's Armor if you really wanted to.
Flat Damage Manmode ENDGAME Build
// These are the only boots that will give you more burst damage, so they are the obvious choice for this build. BUT once you get to super late game, as this build should, you can sell these for a Zephyr as it will give you more burst damage without losing very much movement speed. 
// Early game before your base AD is very high or the enemy team has a lot of HP this item will give you the most burst. 
// This item will give you more overall burst than any other item in the mid game, as long as you are utilizing its passive, but as Lucian you should already be utilized your built in passive Lightslinger which is very similar. So not only is this the most burst item you can get, but it also fits Lucian's playstyle quite nicely. 
// Only 2 items can trump The Bloodthirster for burst damage and you already have them! So now the only thing left to do is get more of these bad boys. 
For a 6th item go for something with survavability. I can't really tell you which item to buy since tanky items are all situational. If you're having trouble against AD a Randuin's Omen is always a good buy. If CC/magic damage is the problem a Mercurial Scimitar is an amazing item. If its a mixture between magic AND physical you could go for a Guardian Angel or even a Warmog's Armor if you really wanted to. BUT this build is supposed to be manmode burst, so you can also opt for ANOTHER The Bloodthirster and do INSANE burst damage.
Overall DPS ENDGAME build
// These boots give the most overall DPS of any boots, so its obvious why we're getting them for this build. Also the reduced cooldowns allow for more frequent poking in the laning phase, and more dashing around in teamfights. Super lategame however you're gonna want to sell these for a Zephyr because it will give you more over DPS without losing much movement speed OR cooldown reduction. 
// Early game before your opponents have much HP this item gives you the most overall DPS of any item. 
// Yes I know the unique passive doesn't stack but you don't need it to. 70 AD and 25% crit brings more Overall DPS to the table than ANY other item besides Blade of the Ruined King. 
For a 6th item go for something with survavability. I can't really tell you which item to buy since tanky items are all situational. If you're having trouble against AD a Randuin's Omen is always a good buy. If CC/magic damage is the problem a Mercurial Scimitar is an amazing item. If its a mixture between magic AND physical you could go for a Guardian Angel or even a Warmog's Armor if you really wanted to. But this build is supposed to be for the most Overall DPS possible and if you really want the most POSSIBLE go with yet ANOTHER Infinity Edge
Item Scores (Based on stat weights below)
As far as the crit items are concerned the crit multiplier I used was 250%, so these are the numbers for if you already own an infinity edge and don't have any other increases to crit damage from Masteries/Runes. With that being said you can see even a SECOND Infinity Edge is much better than a Phantom Dancer. Also if you use the Mastery for increased Crit Damage, Infinity Edge > The Bloodthirster for DPS, but not for DMG.
// Numbers are assuming the target has 2k HP and activate is used on CD. DMG Per Gold  0.348743438671875 DPS Per Gold  0.0709072962984375 DMG  1415.97900375 DPS  231.903348155 
DMG Per Gold  0.3416832531876138 DPS Per Gold  0.0422696051712568 DMG  1313.088742 DPS  162.44209275 
Even without an Infinity Edge this item still does more for overall damage to your abilities than The Bloodthirster, its just very expensive.
// These scores are for a Bloodthirster with max stacks. DMG Per Gold  0.3699054546875 DPS Per Gold  0.05334379659375 DMG  1183.697455 DPS  170.7001491 
DMG Per Gold  0.2606465903947368 DPS Per Gold  0.0439126228276316 DMG  990.4570435 DPS  166.867966745 
DMG Per Gold  0.2137364609649123 DPS Per Gold  0.0390969432105263 DMG  609.14891375 DPS  111.42628815 
DMG Per Gold  0.1812382642857143 DPS Per Gold  0.0395872137321429 DMG  507.46714 DPS  110.84419845 
// Armor Penetration hasn't been factored into this item's score. DMG Per Gold  0.348743438671875 DPS Per Gold  0.0296869824521739 DMG  473.478982 DPS  68.28005964 
When comparing this item to other items, look at how much more damage they do and factor in armor reducing that damage. get rid of 35% of that armor and then factor armor into this items damage. You'll find that this item comes ahead of any item at extreme levels of armor. But sadly only at EXTREME levels. Lucian's AD ratios are just so amazing that this item is lackluster in most late game circumstances.
DMG Per Gold  0.1392109111111111 DPS Per Gold  0.0239958949333333 DMG  125.28982 DPS  21.59630544 
DMG Per Gold  0 DPS Per Gold  0.0221407309125 DMG  0 DPS  22.1407309125 
Stat Weights (Based on maths below)
The numbers for Crit Chance are using 250% crit damage, so crit will only be as good as listed below if you already own an infinity edge.
1 Attack Damage nets you...
11.83697455 more damage and 1.707001491 more DPS.
1% Attack Speed nets you...
6.264491 more damage and 1.079815272 more DPS.
1% Crit Chance nets you...
6.474753 more damage and 1.895114495 more DPS.
1% Cooldown Reduction nets you...
0 more damage and 1.4760487275 more DPS.
As far as pure damage is concerned.
AD>Crit>AS>CDR
As far as overall damage per second is concerned.
Crit>AD>CDR>AS
The Maths (Long and Boring)
Alright, first lets look at Lucian's base stats at level 18. (All math assumes level 18 with all skills maxed and all passive procs are triggered)
AD  97
AS  .994357
Crit  0
Now lets do the math to find out how much damage and DPS Lucian does with his abilities and auto attacks with no items, runes, and masteries. (No bonus AD, which is what every ability scales with except the passive)
Auto Attacks
DMG  97 DPS  97*.994357=96.452629
Passive  Lightslinger
Adds 72.75 damage to each ability, which will be in the below formulas for each ability.
Q  Piercing Light
DMG  240+72.75=312.75 DPS  312.75/5=62.55
W  Ardent Blaze
DMG  220+72.75=292.75 DPS  292.75/10=29.275
E  Relentless Pursuit
DMG  72.75 DPS  7.275
R  The Culling (Shots fired is 7.5+(10.5*.994357)=17.9407485)
DMG  (60*17.9407485)+72.75=1149.19491 DPS  1149.19491/50=22.9838982
Total DMG  97+312.75+292.75+7.275+1149.19491=1858.96991
Total DPS  96.452629+62.55+29.275+7.275+22.9838982=219.5365272
Next up CDR, I'll be using 20% for easier math then later finding out what 1% yields.
Auto Attacks
DMG  97 DPS  97*.994357=96.452629
Passive  Lightslinger
Adds 72.75 damage to each ability, which will be in the below formulas for each ability.
Q  Piercing Light
DMG  240+72.75=312.75 DPS  312.75/4=78.1875
W  Ardent Blaze
DMG  220+72.75=292.75 DPS  292.75/8=36.59375
E  Relentless Pursuit
DMG  72.75 DPS  72.75/8=9.09375
R  The Culling (Shots fired is 7.5+(10.5*.994357)=17.9407485)
DMG  (60*17.9407485)+72.75=1149.19491 DPS  1149.19491/40=28.72987275
Total DPS  96.452629+78.1875+36.59375+9.09375+28.72987275=249.05750175
DPS Difference  249.05750175219.5365272=29.52097455
Now lets find out the DPS difference for 1% CDR.
29.52097455/20=1.4760487275
Now lets add 100 AD and do it all over again to see how much DPS 100 AD will bring us. (100 for easy math later when comparing stats)
New stats with 100 bonus AD
AD  197
AS  .994357
Crit  0
Auto Attacks
DMG  197 DPS  197+.994357=195.888329
Passive  Lightslinger
Adds 147.75 damage to each ability, which will be in the below formulas for each ability.
Q  Piercing Light
DMG  240+120+147.75=507.75 DPS  507.75/5=101.55
W  Ardent Blaze
DMG  220+60+147.75=427.75 DPS  427.75/10=42.775
E  Relentless Pursuit
DMG  147.75 DPS  147.75/10=14.775
R  The Culling (Shots fired is 7.5+(10.5*.994357)=17.9407485)
DMG  ((60+30)*17.9407485)+147.75=1762.417365 DPS  1762.417365/50=35.2483473
Total DMG  197+507.75+427.75+147.75+1762.417365=3042.667365
Total DPS  195.888329+101.55+42.775+14.775+35.2483473=390.2366763
Now lets see how much more DMG/DPS that is exactly.
DMG  3042.6673651858.96991=1183.697455 DPS  390.2366763219.5365272=170.7001491
Now lets divide that by 100 to find a DMG/DPS increase for 1 AD instead of 100.
DMG  1183.697455/100=11.83697455 DPS  170.7001491/100=1.707001491
Now lets do AS, just like before I'll be using 100% AS.
New stats with 100% bonus AS.
AD  97
AS  1.988714
Auto Attacks
DMG  97 DPS  97*1.988714=192.905258
Passive  Lightslinger
Adds 72.75 damage to each ability, which will be in the below formulas for each ability.
Q  Piercing Light
DMG  240+72.75=312.75 DPS  312.75/5=62.55
W  Ardent Blaze
DMG  220+72.75=292.75 DPS  292.75/10=29.275
E  Relentless Pursuit
DMG  72.75 DPS  7.275
R  The Culling (Shots fired is 7.5+(10.5*1.988714)=28.381497
DMG  (60*28.381497)+72.75=1775.63982 DPS  1775.63982/50=35.5127964
Total DMG  97+312.75+292.75+7.275+1775.63982=2485.41482
Total DPS  192.905258+62.55+29.275+7.275+35.5127964=327.5180544
Now lets see how much more DMG/DPS that is exactly.
DMG  2485.414821858.96991=626.4491 DPS  327.5180544219.5365272=107.9815272
Now lets divide that by 100 to find a DMG/DPS increase for 1 AS instead of 100.
DMG  626.4491/100=6.264491 DPS  107.9815272/100=1.079815272
Our final stat will be crit, to make the math really easy on me I'll be using 100% crit.
(For this I'll be using a crit multiplier of 250%, so these numbers for crit will be WITH infinity edge)
New stats with 100% bonus crit
AD  97
AS  .994357
Crit  100
Auto Attacks
DMG  242.5 DPS  242.5*.994357=241.1315725
Passive  Lightslinger
Adds 181.875 damage to each ability, which will be in the below formulas for each ability.
Q  Piercing Light
DMG  240+181.875=421.875 DPS  421.875/5=84.375
W  Ardent Blaze
DMG  220+181.875=401.875 DPS  401.875/10=40.1875
E  Relentless Pursuit
DMG  181.875 DPS  181.875/10=18.1875
R  The Culling
DMG  (60*17.9407485)+181.875=1258.32021 DPS  1258.32021/50=25.1664042
Total DMG  242.5+421.875+401.875+181.875+1258.32021=2506.44521
Total DPS  241.1315725+84.375+40.1875+18.1875+25.1664042=409.0479767
Now lets see how much more DMG/DPS that is exactly.
DMG  2506.445211858.96991=647.4753 DPS  409.0479767219.5365272=189.5114495
Now lets divide that by 100 to find a DMG/DPS increase for 1 Crit instead of 100.
DMG  647.4753/100=6.474753 DPS  189.5114495/100=1.895114495
This next section is concerning the Blade of the Ruined King's passive. For numbers I'm going to assume the target has 2k HP then find out how much overall DMG and DPS the passive alone will give you, so I can use it to more properly find its value.
Auto Attacks
DMG  97+100=197 DPS  197*.994357=195.888329
Passive  Lightslinger
Adds 172.75 damage to each ability, which will be in the below formulas for each ability.
Q  Piercing Light
DMG  240+172.75=415.75 DPS  83.15
W  Ardent Blaze
DMG  220+172.75=393.75 DPS  393.75/10=39.275
E  Relentless Pursuit
DMG  172.75 DPS  172.75/10=17.275
R  The Culling (Shots fired is 7.5+(10.5*.994357)=17.9407485)
DMG  (60*17.9407485)+172.75=1249.19491 DPS  1249.19491/50=24.9838982
Blade of the Ruined King ACTIVATE
DMG  300 DPS  300/60=5
Total Damage  2728.44491
Total DPS  365.5722272
BONUS Damage  2728.444911858.96991=869.475
BONUS DPS  365.5722272219.5365272=146.0357
Now we'll test Muramana's Active toggle with full mana equal to 1927.
Auto Attacks
DMG  97 DPS  97*.994357=96.452629
Passive  Lightslinger
Adds 72.75 damage to each ability, which will be in the below formulas for each ability.
Q  Piercing Light
DMG  240+72.75+115.62=428.37 DPS  428.37/5=85.674
W  Ardent Blaze
DMG  220+72.75+115.62=408.37 DPS  408.37/10=40.837
E  Relentless Pursuit
DMG  72.75 DPS  72.75/10=7.275
R  The Culling
DMG  (60*17.9407485)+72.75+115.62=1264.81491 DPS  1264.81491/50=25.2962982
Total DMG  2271.66491
Total DPS  255.5349272
BONUS DMG  412.695
BONUS DPS  35.9984
Lastly I'll be testing Trinity Force's AP and passive. I'm grouping the ability power with Trinity Force because I don't plan on making an AP so it's not very important I'm just being thorough figuring out just how good Trinity Force is.
Auto Attacks
DMG  97 DPS  97*.994357=96.452629
Passive  Lightslinger
Adds 72.75 damage to each ability, which will be in the below formulas for each ability.
Q  Piercing Light
DMG  240+72.75+145.5=438.25 DPS  438.25/5=87.65
W  Ardent Blaze
DMG  220+72.75+27+145.5=465.25 DPS  465.25/10=46.525
E  Relentless Pursuit
DMG  72.75+145.5=218.25 DPS  218.25/10=21.825
R  The Culling (Shots fired is 7.5+(10.5*.994357)=17.9407485)
DMG  (63*17.9407485)+72.75+145.5=1348.5171555 DPS  1348.5171555/50=26.97034311
Total DMG  2564.2671555
Total DPS  279.42297211
Bonus DMG  705.2972455
Bonus DPS  59.88644491
Summary (The End!)
Thanks for checking out my builds, before voting or commenting please at least try them or check out the math that indisputably PROVES these builds to be the best in their own category. Some of the results were actually very surprising to me, I was expecting crit to be very strong, but it turns out Lucian's crazy AD ratios (Mainly dat Passive 75% on EVERY ability) just overpowers everything else. Theres been a lot of speculation that the classic ADC builds should also be used on Lucian and although they are good and will WORK the builds in this guide are 100% guaranteed to be the best possible. According to math at least... // SPECIAL THANKS: Credits to jhoijhoi for the template, which you can find here. 

