Generalized Fibonacci Progression

[mr.ore]

I got idea how to generalize classical Fibonacci progression on any kind of chance, and there is the result:

Let's have a series of numbers defined like this:

[tex]X_{n}^{p}=\left\{\begin{array}{rll}n\leq0&:&0\\n=1&:&1\\n > 1&:&\left\lceil \frac{1}{p}\sum_{k=0}^{\lfloor p \rfloor}{X_{n-1-k}^{p}}\right\rceil\end{array}[/tex]

where [tex]p[/tex] is payout of given chance, and we suppose that payout is not much different from event's probability. It is not power, I put it to that place where the power of some value usually is to indicate, that it is for event with payout p.

We start play with [tex]X_1^p[/tex] and if lost, add another number to right of the series given by the formula above. If we win, remove [tex]1+\lfloor p \rfloor[/tex] numbers from the right of the series.

Let's look how the series looks for payout p=1:1 = 1:
[tex]X_n^1=\left\lceil \frac{1}{1} \left(X_{n-1}^{1}+X_{n-2}^{1}\right) \right\rceil[/tex]
and it leads to series
[tex]1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,...[/tex]
which is classical Fibonacci series, and on win we cross off 1+floor(1) = 2 numbers from right, floor = round number down. This is classical Fibonacci progression, which is special case of the general idea that one win cancels some previsous loses according to payout and probability so that we in the end cancel out the series.

How would this look on dozen/column chances? p=2:1=2:
[tex]X_n^2=\left\lceil \frac{1}{2} \left(X_{n-1}^{2}+X_{n-2}^{2}+X_{n-3}^{2}\right) \right\rceil[/tex]
and the series is
[tex]1,1,1,2,2,3,4,5,6,8,10,12,15,19,23,29,36,44,55,68,84,104,128,158,195,241,297,367,\\453,559,690,851,1050,1296,1599,1973,2434,3003,3705,4571,5640,...[/tex]
On win we cross 1 + floor(2) = 3 numbers from right.

Line p=5:1=5:
[tex]X_n^5=\left\lceil \frac{1}{5} \left(X_{n-1}^{5}+X_{n-2}^{5}+X_{n-3}^{5}+X_{n-4}^{5}+X_{n-5}^{5}+X_{n-6}^{5}\right) \right\rceil[/tex]
and the series is

[tex]1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,7,8,8,9,9,10,11,11,12,13,14,15,16,17,18,19,20,21,23,24,25,\\27,28,30,32,34,36,38,40,42,45,47,50,53,56,59,62,66,70,74,78,82,87,92,97,102,108,114,120,127,134,\\141,149,157,166,175,185,195,206,217,229,242,255,269,284,300,316,334,352,371,392,413,436,460,\\485,512,540,...[/tex]
On win we cross 1 + floor(5) = 6 numbers from right.

Corner p=8:1=8:
[tex]X_n^8=\left\lceil \frac{1}{8} \left(X_{n-1}^{8}+X_{n-2}^{8}+X_{n-3}^{8}+X_{n-4}^{8}+X_{n-5}^{8}+X_{n-6}^{8}+X_{n-7}^{8}+X_{n-8}^{8}+X_{n-9}^{8}\right) \right\rceil[/tex]
and the series is
[tex]1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,\\8,8,8,8,9,9,9,10,10,10,11,11,11,12,12,12,13,13,14,14,14,15,15,16,16,17,17,18,18,19,19,\\20,20,21,22,22,23,23,24,25,25,26,27,28,28,29,30,31,32,32,33,34,35,36,37,38,39,40,41,\\42,43,44,45,47,48,49,50,52,53,54,56,57,59,60,62,63,65,67,68,70,72,74,76,78,80,82,84,\\86,88,90,93,95,97,100,102,105,107,110,113,116,119,122,125,128,131,134,138,141,145,\\148,152,156,160,164,168,172,176,181,...[/tex]
On win we cross 1 + floor( 8 ) = 9 numbers from right.

Street p=11:1=11:
[tex]X_n^{11}=\left\lceil \frac{1}{11} \left(X_{n-1}^{11}+X_{n-2}^{11}+X_{n-3}^{11}+\cdots+X_{n-10}^{11}+X_{n-11}^{11}+X_{n-12}^{11}\right) \right\rceil[/tex]
and the series is
[tex]1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,\\5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,\\11,11,11,11,11,12,12,12,12,13,13,13,13,14,14,14,14,15,15,15,15,16,16,16,\\17,17,17,17,18,18,18,19,19,19,20,20,20,21,21,21,22,22,22,23,23,24,24,24,\\25,25,26,26,26,27,27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35,35,36,36,\\37,38,38,39,39,40,41,41,42,42,43,44,44,45,46,46,47,48,49,49,50,51,52,52,53,\\54,55,56,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,\\78,80,81,82,83,84,85,87,88,89,90,92,93,94,96,97,98,100,101,...[/tex]
On win we cross 1 + floor( 11 ) = 12 numbers from right.

Split p=17:1=17:
[tex]X_n^{17}=\left\lceil \frac{1}{17} \left(X_{n-1}^{17}+X_{n-2}^{17}+X_{n-3}^{17}+\cdots+X_{n-16}^{17}+X_{n-17}^{17}+X_{n-18}^{17}\right) \right\rceil[/tex]
and the series is
[tex]1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,\\4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,\\8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,\\12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,\\16,16,16,16,16,16,16,17,17,17,17,17,17,17,18,18,18,18,18,18,19,19,19,19,19,19,20,20,20,20,20,20,\\21,21,21,21,21,21,22,22,22,22,22,23,23,23,23,23,23,24,24,24,24,24,25,25,25,25,25,26,26,26,26,26,\\27,27,27,27,27,28,28,28,28,29,29,29,29,29,30,30,30,30,31,31,31,31,32,32,32,32,33,33,33,33,\\34,34,34,34,35,35,35,35,36,36,36,36,37,37,37,38,38,38,38,39,39,39,40,40,40,40,41,41,41,\\42,42,42,43,43,43,43,44,44,44,45,45,45,46,46,46,47,47,47,48,48,48,49,49,49,50,50,50,\\51,51,51,52,52,53,53,53,54,54,54,55,55,56,56,...[/tex]
On win we cross 1 + floor( 17) = 18 numbers from right.

Straight up p=35:1=35:
[tex]X_n^{35}=\left\lceil \frac{1}{35} \left(X_{n-1}^{35}+X_{n-2}^{35}+X_{n-3}^{35}+\cdots+X_{n-16}^{34}+X_{n-35}^{35}+X_{n-36}^{35}\right) \right\rceil[/tex]
and the series is
[tex]1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,\\2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,\\3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,\\4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,\\5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,\\6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,\\7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,\\8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,\\9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,\\10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,\\11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,\\12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,\\13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,\\14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,\\15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,\\16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,\\17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,\\18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,\\19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,\\20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,\\21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,\\22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,\\23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,\\24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,\\25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,\\26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,\\27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,\\28,28,28,28,28,28,28,28,28,28,28,28,28,28,28,\\29,29,29,29,29,29,29,29,29,29,29,29,29,29,\\29,30,30,30,30,30,30,30,30,30,30,30,30,30,30,\\31,31,31,31,31,31,31,31,31,31,31,31,...[/tex]

On win we cross 1 + floor( 35) = 36 numbers from right.


That's it.

It is possible to convert one such line to another, you just sum up your line and get total loss, and then you start making new line until it's sum is greater or equal to your loss. Then you got your new line, and you play on another chance.

It is also possible to make it work on eg. n numbers, but it will need little hack, you will sometimes cross 1+floor(p) numbers but from time to time you will have to cross 1+ceil(p) numbers because p is not integer, and you have to make up for it. This will be explained later, now I'm coding it, just for fun.

All these methods are equivalent, average gain per spin is same, and you can lost with any of them. But it's possible to convert one progression to another, so we can try to sufr dispersion with it. If our bet seems too big, we can convert series to one with higher payout, and play on it until we decide to go back to higher level. Or it can be done in reverse, if we'are winning we play on high payout, and on losing we go down, thus increasing our hit rate. Neither of those I tested yet, but idea seems interesting.

[mr.ore]

For some unknown reason there is the title of this post "Generalized Fibonacci Progression" after each latex section. It is not in preview, even when modifying. Does anyone know how to remove it?

[mr.ore]

And now some test result, which proved obvious. This method is no different from original Fibonacci series, in the end, we end up in a big hole. Only "advantage" is that we will not hit limits so fast as with martingale. There are in this order:
dozen, line, corner, street, split, straight. 4096 spins is nothing much, but enough to see, what it could do.

[mr.ore]

More spins on single number...

[mr.ore]

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