Method to obtain the decimal expansion of 1/801 using the fibonacci sequence - 1.

[Davide Rotondo]

Hi to all dear seq fans, I want to show you this method I found similar to that with is used to obtain 1/89 using Fibonacci numbers.

Method to obtain the decimal expansion of 1/801 using the fibonacci sequence - 1.

1/801 = 0.001248439450686641697877652933832...

Fibonacci(n) - 1 = 0, 0, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583, 4180, 6764, 10945, 17710, 28656, 46367, 75024, ...

METHOD
0
00
0 0 1
0 0 0 2
0 0 0 0 4
0 0 0 0 0 7
0 0 0 0 0 1 2
0 0 0 0 0 0 2 0
0 0 0 0 0 0 0 3 3
0 0 0 0 0 0 0 0 5 4
0 0 0 0 0 0 0 0 0 8 8
0 0 0 0 0 0 0 0 0 1 4 3
. . .
Adding the values ​​of the columns with the appropriate carryovers, we obtain the decimal expansion of 1/801.

[Davide Rotondo]

Another curious strange thing is that if we take in to consideration Fibonacci + 1 with this method we get:  1,2,2,3,4,7,0,6,6,1,6,7,2,9,...

Helped by the matching system of OEIS I noticed that this is deltas matching: a(n+1) - a(n) = 1, 0, 1, 1, 3, 7, 6, 0, 5, 5, 1, 5, 7

• A021805 Decimal expansion of 1/801.
  (8, 7, 7, 6, 5, 2, 9, 3, 3, 8, 3, 2, 7, 0)

The curios thing is that the match start from  0, 0, 1, 2, 4, 8, 4, 3, 9, 4, 5, 0, 6, 8, 6, 6, 4, 1, 6, 9, 7, 8, 7, 7, 6, 5, 2, 9, 3, 3, 8, 3, 2, 7, 0, 9, 1, 1, 3, 6, 0, 7, 9, 9, 0, 0, 1, 2, 4, 8, 4, 3, 9, 4, 5, 0, 6, 8, 6, 6, 4, 1, 6, 9, 7, 8, 7, 7, 6, 5, 2, 9, 3, 3, 8, 3, 2, 7, 0, 9, 1, 1, 3, 6, 0, 7, 9, 9, 0, 0, 1, 2, 4, 8, 4, 3, 9, 4, 5

What do you think?

Davide

[Kevin Ryde]

Davide Rotondo <david...@gmail.com> writes:
>
> 1/89 using Fibonacci numbers.

The key is substituting x = 1/10 into the generating function
x/(1-x-x^2) of the Fibonacci numbers. The effect is your
adding at decimal positions.

All linear recurrences have a similar substitution to make a
rational. If the gf numerator is 1 or x or otherwise simple
it can make "1/something".

> 1/801 using the fibonacci sequence - 1.

The -1*(1/10)^n is a geometric sum, or think of it as modifying the
generating function, so, umm, 10/89 - 1/9.

> Adding the values of the columns with the appropriate carryovers,

If you stop at a certain Fibonacci number, and without each "-1",
you can notice terms of A094704.

[Davide Rotondo]

Thanks a lot Kevin. 

What about the second mail (the Deltas match)?

Regards 

Davide

--
You received this message because you are subscribed to the Google Groups "SeqFan" group.
To unsubscribe from this group and stop receiving emails from it, send an email to seqfan+un...@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/seqfan/87344lhafu.fsf%40blah.blah.

[Davide Rotondo]

Also works for 1/979 using Fibonacci numbers +1 intersperses with Fibonacci numbers -1.

What do you think? Can we arrive at a general rule?

1/979=0.001021450459652706843718...
Fibonacci + 1 = 1, 2, 2, 3, 4, 6, 9, 14, 22, ...
Fibonacci -1 = -1, 0, 0, 1, 2, 4, 7, 12, 20, ...
METHOD
1

0 0
0 0 2

0 0 0 1

0 0 0 0 4

0 0 0 0 0 4
0 0 0 0 0 0 9
0 0 0 0 0 0 1 2
0 0 0 0 0 0 0 2 2

0 0 0 0 0 0 0 0 3 3

0 0 0 0 0 0 0 0 0 5 6
0 0 0 0 0 0 0 0 0 0 8 8
. . .
Adding the values ​​of the columns with the appropriate carryovers, we obtain the decimal expansion of 1/979.

Davide

[Kevin Ryde]

Davide Rotondo <david...@gmail.com> writes:
>
> What about the second mail (the Deltas match)?

+1 or -1 in the terms becomes +/- 1/9 = 0.111... or 1/90 = 0.0111...
in the rational. You could check whether your change makes for
instance "(1/801 - something) * 10^12" which would explain.

Decimals of fractions X/801 can have a few different periodic parts
so can look for why didn't switch across to a different one.

[Stephen Lucas]

Greetings, all. I’ve had a chance to look at Davide’s new results, and can explain them using generating functions again. 

First let g_n=f_n-1, one less than Fibonacci, with g_1=0, g_2=0, g_3=1. This gives g_n=g_{n-1}+g_{n-2}+1. To get rid of the +1 on the right hand side, subtract g_{n-1}=g_{n-2}+g_{n-3}+1, giving g_n=2*g_{n-1}+g_{n-3}. This recurrence relation with given initial conditions has generating function G(x)=x^3/(1-2*x+x^3), putting x=1/10 gives the fraction 1/801, as found experimentally.

If we look at one more than Fibonacci, we get exactly the same 3rd order recurrence relation, but this time with initial values 1,2,2 (if we start from the zeroth term this time to match what Davide did). This time the generating function is (x-2*x^3)/(1-2*x+x^3) which leads to the fraction 98/801, which matches.

Finally, Davide noticed a result by alternating Fibonacci -1 and +1. We can explain this because both cases have the same recurrence relation. The alternating spreads out coefficients, so if we start with g_n=2g_{n-2}+g_{n-6} with 6 initial conditions given by alternating the above results, we should get a generating function that leads to the fraction. In principle, the same trick would work for other sequences, if there is a reason to do so.

Steve

--
You received this message because you are subscribed to the Google Groups "SeqFan" group.
To unsubscribe from this group and stop receiving emails from it, send an email to seqfan+un...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/seqfan/98c39c7a-5dbf-4726-ad18-c4ba9b0b5668n%40googlegroups.com.