The saga of the Luke/Audi Fibonacci Relation

To make a long story short, Mr. FibonacciCoin came in and dropped a bomb on #woodcoin IRC the other day.

This story is simply far too big to do anything other than completely drop the thing at your feet as it went down.  Yes I know this isn't the traditional method of academic aggrandizement but watcha gonna do?  This is just a blog after all, more official publication is hardly ruled out by the existence of these chicken scratchings.

02:20     FibonacciCoin     8 is a great number
02:20     funkenstein_     no five in that though!
02:21     FibonacciCoin     5 and 3
02:21         🙂
02:21     funkenstein_     ah..  i always tend to think prime decomposition, gotta remember addition too
02:21     FibonacciCoin     3 times for emphasis
02:22     funkenstein_     so you could simply decrease the block reward by 0.618..  every cycle
02:23         cycle length could go up with fibonnaci series
02:23     FibonacciCoin     This kinda interests me.. 221-5 332-8 553-13 885-21 13138-34 212113-55.. why does this work that way
02:24         I thought of that..
02:24     funkenstein_     hey that's neat
02:25     FibonacciCoin     Two fibonacci numbers and the previous one together equal the next one..
02:25     funkenstein_     2f_(n)+f_(n-1)=f_(n+2)

You follow that so far?  In his series demonstration notation the left hand side has implied summation of fibonacci numbers so 212113-55 is saying:

21+21+13=55.

My formula, phrasing his discovery in more conventional terms, also looks like

f_(n)+2*f_(n+1)=f_(n+3)

It's a bit like the ordinary generator for the fibonacci series

f_(n)+f_(n+1)=f_(n+2)

except we've shifted predicted term out further.  Cool huh?  Brace yourself, it gets way way cooler.

OK so on day two we had this discussion:

FibonacciCoin     I meant it was an interesting post
00:24     GoldenAngle     I was up thinking last night about 122=5 233=8 355=13  etc. and suddenly it hit me it works with two and 3 and 5 and 8 and 13 numbers too! for example 11222=8 22333=13 33555=21 55888=34 & 11122222=13 222555=21 33355555=34 55588888=55 & 1111122222222=21 2222233333333=34 3333355555555=55 5555588888888=89 & 111111112222222222222=34 or 2x8+3x13=55 3x8+5x13=89 5x8+8x13=144 etc. I dont know how to put this in an algebraic equation but this really
00:24         blows mhy mind.
00:25         EUREKA!

Indeed.. EUREKA!  This thing is awesome.  Anyway I'm going to include the next one hour of conversation as this bomb was slowly digested.  This is rather unconventional reveal as it bares my then clear ignorance of the importance of this relation.  Give us a break, it was irc - we were doing other things.

00:28     funkenstein_     howdy
00:30         GoldenAngle, I didn't see your formula in the list on wikipedia of recurrence relations for Fibonacci series
00:31         thanks whaack 🙂
00:32         we had f_n+2*f_(n-1)=f_(n+2)
00:33         hmm that is cool...  tricky to symbolize
00:33         nice one!!
00:35     GoldenAngle     I feel like a genius!
00:35         Im fibonaccicoin by the way
00:36         Can this somehow be used to fractionalize the fibonacci sequence is my question.
00:36         This has to be important somehow
00:41         Im digging for another relationship. There has to be one
00:42     funkenstein_     fractionalize?
00:43         i haven't seen a recurrence relation in which the coeficients are determined by the function in question bofore
00:43     GoldenAngle     ya can it work with 0 and negative numbers and part of 1
00:43     funkenstein_     i'm no number theory guy though
00:44     GoldenAngle     Did I discover something perhaps?
00:44     funkenstein_     even the first relation I didn't turn up in my cursory search
00:44     GoldenAngle     I just think there has to be something to this
00:44     funkenstein_     lets write it up 🙂
00:45     GoldenAngle     I agree I need to publish my findings somehow
00:45         If you could help I would greatly appreciate it
00:45     funkenstein_     and pay some guy with a low erdös number to be a coauthor
00:45     GoldenAngle     absolutely
00:45     funkenstein_     hehe
00:46     GoldenAngle     I'm so excited
00:46     funkenstein_     nice
00:46         And I just can't hide it
00:47     GoldenAngle     I can't believe it.. the first relationship came to me like 2 years ago.. and I just sat up at night thinking about it drawing a blank for 2 years. then last night
00:49     funkenstein_     perhaps it's just
00:50         F_n*F_m + F_(n+1)*F_(m+1) = F_(n+m+1)
00:51     GoldenAngle     is that it?
00:51     funkenstein_     check it for me couldya?
00:54     GoldenAngle     if i were good at algebra i would
00:56         whaack joined #woodcoin
00:57     GoldenAngle     its simple enough that it seems right to me
00:57     funkenstein_     come on now you figured out those lists
00:57         just plug in some examples
01:00         "the only cubic fibonacci numbers are 1 and 8"
01:01         d'Ocagne's identity  <-- similar kind of thing but not the same
01:01     GoldenAngle     thats really interesting
01:02     funkenstein_     http://mathworld.wolfram.com/FibonacciNumber.html
01:02     funkshelper     [ Fibonacci Number -- from Wolfram MathWorld ] - mathworld.wolfram.com
01:02     GoldenAngle     I wanna name it after my Baby luke who was born last month and had spina bifida and might not walk if I can pick the name 🙂
01:03     funkenstein_     sending good luck his way now
01:03         are you familiar with the Lucas numbers?
01:04     GoldenAngle     yes I was gonna mention that
01:04         I was thinking part of a fibonacci coin could be lucas' or Luke
01:04         My middle name's Luke 🙂
01:05         I was thinking about your coin too last night
01:05         had some ideas.
01:06     funkenstein_     I'm all ears
01:06     GoldenAngle     Slinging truckloads of wood and giving people a discount if they used LOGS to purchase them..
01:07         Lose a little bit of margin but make up for it in getting the coin utilized and popularized as a commodity trade coin
01:07     funkenstein_     I had considered that long ago, I approached a few merchants who already did bitcoin and some wood products
01:07     GoldenAngle     The way you could force the use is by dealing directly with the consumer
01:08     funkenstein_     generally you don't want to force the customer to use it, but allowing use w/ discount ..
01:08     GoldenAngle     You can't force the hand of a supplier but consumers are willing to do anything for small discount
01:08     funkenstein_     ^
01:08     GoldenAngle     right i don't mean force.. I mean encourage strongly
01:08     funkenstein_     and supplier will be happy to take a small bonus too on conversion
01:08     GoldenAngle     🙂
01:08         I have lots of experience with this
01:08     funkenstein_     well you might know the right people to contact
01:09         hmm seems like you are one step ahead of me in this conversation 🙂
01:09     GoldenAngle     I used to import container loads from Brazil and China and sell direct to consumer
01:09     funkenstein_     that is IDEAL application
01:09     GoldenAngle     I even had distributors and manufacturers buy from me when they were low on stock
01:09     funkenstein_     because intl trade is frictionless with coin
01:10     GoldenAngle     yea everything was wiretransfer when i bought
01:10         bill of lading
01:10     funkenstein_     i'm sure a lot of these folks will need somebody to hold their hand through the process
01:11         and will consider the "end" to be a private bank account balance
01:11         but that's no problem
01:11         anyway I should keep my mouth shut you know a lot more than me here
01:12         how can I encourage you to research further?
01:12         oddly Abramowitz&Stegun seems to have no reference to Fibonacci numbers
01:12     GoldenAngle     right.. i was thinking you could start with getting your volumes of coin trade by starting small with a few thousand dollar purchases and work up to imports
01:13         You don't need to keep your mouth shut. your obviously a very smart person. Im appreciative for your listening actually
01:14     funkenstein_     well all I can really provide here is a bit of LOG as incentive
01:14     GoldenAngle     I will gladly research further.. help me publish my fibonacci findings to gain credence to my coin idea maybe. I'd love to work together somehow
01:14     funkenstein_     thanks as well 🙂
01:14         excellent, on it 🙂
01:16     GoldenAngle     I'll be pondering on how to improve woodcoin
01:17         I have a solid reputation in the wood flooring trade market. I'll post my trademark picture. you are free to use it on your website. I only accept log for payment on my selling of hardwood flooring.
01:18     funkenstein_     wow awesome
01:18     GoldenAngle     I'd be a good referral.
01:18     funkenstein_     i think i'm gonna get my floors redone 🙂
01:18     GoldenAngle     lol
01:18     funkenstein_     that formula seems to work by the way
01:19     GoldenAngle     awesome
01:21     funkenstein_     the Luke Recursive Fibonacci Relation
01:22     funkenstein_     nice logo 🙂
01:22     GoldenAngle     Luke Audi - son/daughter
01:23         Thanks Im sure I can get a better image for you.. I'll find one.
01:23     funkenstein_     without the 1 at the end
01:23         it's just
01:23         F_n*F_m + F_(n+1)*F_(m+1) = F_(n+m)
01:24     GoldenAngle     Your awesome I really appreciate it.
01:24     funkenstein_     somebody probably wrote it up somewhere
01:24     GoldenAngle     But if Not!
01:24     funkenstein_     but it's definitely cool, great find in any case :0
01:25     GoldenAngle     Where should I publish it?
01:26     funkenstein_     I think it's novel.  Such a simple form suggests it would be lying around in the usual places
01:26     GoldenAngle     Im wondering if my business partner Kevin Country Flooring Direct would be open to doing some LOG trading.
01:27         I know.. It's so simple.. but it really could have some implications I think
01:27     funkenstein_     d'Ocagne's identy is similar but not as elegant
01:27         the thing is, generally coins that try to have niche uses are kinda missing the point
01:29         but even so, I'd be interested 🙂  if it's helpful to the people involved
01:29     GoldenAngle     I hear you
01:33         IF ive stumbled on something big I certainly won't forget your contribution
01:33         Im fortunate to have meet you
01:34     funkenstein_     send me an email at funkenstein@woodcoin.org
01:35         we got it logged here
01:36         all went down on #woodcoin

  Looks like this thing exists in the literature in a shifted form.  Surprise surprise, nothing new under the sun.  Still very cool though.  I'm following up some loose ends.  Here it is with proper +1 at the end, as per my original suggestion (not the broken later one):

$F_n*F_m + F_{n+1}*F_{m+1} = F_{n+m+1}$

Or for those of you whose adblockers don't like my Latex displayer:

F_n*F_m + F_{n+1}*F_{m+1} = F_{n+m+1}