# [seqfan] Re: Fibonacci numbers in Golden Ratio Base

Neil Sloane njasloane at gmail.com
Wed Oct 15 21:22:55 CEST 2014

```Dale, those are fascinating videos!

I also added them to the Index to the OEIS in the entry
for "videos, sequences with ..."

They really enhance the OEIS and I wish
we had more videos and movies and animations.

If you make any more videos that we can
use to illustrate sequences,

Best regards

Neil

On Tue, Oct 14, 2014 at 7:09 PM, Dale Gerdemann <dale.gerdemann at gmail.com>
wrote:

> 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
>
> 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:
> silly idea that these wave patterns (real or imagined) might be useful for
> generating some fractals: https://www.youtube.com/watch?v=mFr0BQj2mMo and
>
> 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!
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

--
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.