[seqfan] Re: Fibonacci numbers in Golden Ratio Base

[Dale Gerdemann]

Hello Gordon, Hello SeqFans,

I'm glad you like it. Ye, I used a greedy algorithm, but not what you
probably think. To get the golden ratio base representation of an integer
m, I get a greedy solution for m*f_n = f_{n+a} + f_{n+b} + ... + f_{n-y} +
f_{n-z}, where f_n is the nth combinatorial Fibonacci number (f_0=1,
f_1=1) and n is a number somewhere around 40 to 100. After solving this
equation, I reset n to zero, making m*f_n = m.

I got the idea for this algorithm from p. 15 of Proofs That Really Count by
Benjamin and Quinn. According to Benjamin and Quinn, this is only valid
when the value substituted for n is large enough to keep the smallest
indices (f_{n-y} + f_{n-z}) from going negative. I think this can be
interpreted to mean that Benjamin & Quinn's tile counting proofs only work
when n is large enough. I have a paper in the Fibonacci Quarterly, in which
I use a different, less practical algorithm (Combinatorial proofs of
Zeckendorf family identities, 2008/2009).

Another interesting pattern is for Golden Rectangle Numbers A001654. I made
a short video illustrating this pattern, along with other columns of the
Fibonomial Triangle A010048: https://www.youtube.com/watch?v=1LtjGV-nLG0.

Also interesting are the two bisections of the Lucas numbers A005248 (digit
minimizer) and A002878 (digit maximizer). I particularly like the multiples
of A005248 because I have this image of the two digits piling up on top of
each other and then spreading out like waves:
https://www.youtube.com/watch?v=EQYQ4bEUX34. I had the probably rather
silly idea that these wave patterns (real or imagined) might be useful for
generating some fractals: https://www.youtube.com/watch?v=mFr0BQj2mMo and
https://www.youtube.com/watch?v=CnSzunIFnlY.

Do you have a favorite sequence that you'd like to see in golden ratio base
(or maybe Zeckendorf or Bunder (using only negative indexed Fibonacci
numbers) representation?

Dale


On Tue, Oct 14, 2014 at 7:07 PM, Gord! <gord at mathpickle.com> wrote:

> Dale - this is a very attractive pattern.  I presume it is generated
> greedily.  Do you have an explanation for many of the patterns?  What a
> great sequence for school children.
>
> Gord!
>
> On 12 October 2014 11:11, Dale Gerdemann <dale.gerdemann at gmail.com> wrote:
>
> > Hello Seqfans,
> >
> >
> > In the table below, i've listed the powers of phi used in golden ratio
> base
> > to represent the Fibonacci numbers. So, for example, 5 = phi^3 + phi^-1 +
> > phi^-4 = f(3)+f(-1)+f(-4) [where f(n) = Fibonacci(n+1), generalized to
> > negative indices using "precursion"].
> >
> > The pattern strikes me as remarkable. Maybe deserving a place in the
> OEIS?
> >
> >
> > [0] 1
> >
> > [0] 1
> >
> > [1, -2] 2
> >
> > [2, -2] 3
> >
> > [3, -1, -4] 5
> >
> > [4, 0, -4] 8
> >
> > [5, 1, -3, -6] 13
> >
> > [6, 2, -2, -6] 21
> >
> > [7, 3, -1, -5, -8] 34
> >
> > [8, 4, 0, -4, -8] 55
> >
> > [9, 5, 1, -3, -7, -10] 89
> >
> > [10, 6, 2, -2, -6, -10] 144
> >
> > [11, 7, 3, -1, -5, -9, -12] 233
> >
> > [12, 8, 4, 0, -4, -8, -12] 377
> >
> > [13, 9, 5, 1, -3, -7, -11, -14] 610
> >
> > [14, 10, 6, 2, -2, -6, -10, -14] 987
> >
> > [15, 11, 7, 3, -1, -5, -9, -13, -16] 1597
> >
> > [16, 12, 8, 4, 0, -4, -8, -12, -16] 2584
> >
> > [17, 13, 9, 5, 1, -3, -7, -11, -15, -18] 4181
> >
> > [18, 14, 10, 6, 2, -2, -6, -10, -14, -18] 6765
> >
> > [19, 15, 11, 7, 3, -1, -5, -9, -13, -17, -20] 10946
> >
> > [20, 16, 12, 8, 4, 0, -4, -8, -12, -16, -20] 17711
> >
> > [21, 17, 13, 9, 5, 1, -3, -7, -11, -15, -19, -22] 28657
> >
> > [22, 18, 14, 10, 6, 2, -2, -6, -10, -14, -18, -22] 46368
> >
> > [23, 19, 15, 11, 7, 3, -1, -5, -9, -13, -17, -21, -24] 75025
> >
> > [24, 20, 16, 12, 8, 4, 0, -4, -8, -12, -16, -20, -24] 121393
> >
> > [25, 21, 17, 13, 9, 5, 1, -3, -7, -11, -15, -19, -23, -26] 196418
> >
> > [26, 22, 18, 14, 10, 6, 2, -2, -6, -10, -14, -18, -22, -26] 317811
> >
> > [27, 23, 19, 15, 11, 7, 3, -1, -5, -9, -13, -17, -21, -25, -28] 514229
> >
> > [28, 24, 20, 16, 12, 8, 4, 0, -4, -8, -12, -16, -20, -24, -28] 832040
> >
> > [29, 25, 21, 17, 13, 9, 5, 1, -3, -7, -11, -15, -19, -23, -27, -30]
> 1346269
> >
> > [30, 26, 22, 18, 14, 10, 6, 2, -2, -6, -10, -14, -18, -22, -26, -30]
> > 2178309
> >
> > [31, 27, 23, 19, 15, 11, 7, 3, -1, -5, -9, -13, -17, -21, -25, -29, -32]
> > 3524578
> >
> > [32, 28, 24, 20, 16, 12, 8, 4, 0, -4, -8, -12, -16, -20, -24, -28, -32]
> > 5702887
> >
> > [33, 29, 25, 21, 17, 13, 9, 5, 1, -3, -7, -11, -15, -19, -23, -27, -31,
> > -34] 9227465
> >
> > [34, 30, 26, 22, 18, 14, 10, 6, 2, -2, -6, -10, -14, -18, -22, -26, -30,
> > -34] 14930352
> >
> > [35, 31, 27, 23, 19, 15, 11, 7, 3, -1, -5, -9, -13, -17, -21, -25, -29,
> > -33, -36] 24157817
> >
> > [36, 32, 28, 24, 20, 16, 12, 8, 4, 0, -4, -8, -12, -16, -20, -24, -28,
> -32,
> > -36] 39088169
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
>
> --
> Gordon Hamilton
> MMath, PhD
>
> www.MathPickle.com
> Put your students in a pickle!
>
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>