[seqfan] Re: Sequences related to Fibonacci representations

[Charles Greathouse]

If after reasonable consideration you can't find a different way to define
them, just submit them regardless. The sequences are presumably interesting
(coming from Carlitz) so just do your best.

If you do have a conditional simplification it's probably worth mentioning
in the comments.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Mon, Sep 15, 2014 at 3:45 AM, Eric Schmidt <eric41293 at comcast.net> wrote:

> Thanks Olivier and Neil for your replies.
>
> It's been about a month, and I still haven't added the missing sequences
> (15 of them) from Carlitz. I have been putting this off because I don't
> know how to deal with the "chase sequence" issue that came up with the
> already existing sequences. I agree that this is undesirable, but the chase
> is built into the definitions in the paper, with many functions defined in
> terms of previously defined functions. In some cases a theorem in the paper
> could maybe reduce the depth/breadth of the chase, but it doesn't look
> entirely unavoidable to me.
>
> I hope to finish this off in the next few days, so advice on this would be
> appreciated.
>
>
> On 8/17/2014 1:15 AM, Eric Schmidt wrote:
>
>> Recently I came across some sequences whose names were simply "Related
>> to Fibonacci representations".
>>
>> https://oeis.org/search?q=name%3A%22related+to+
>> fibonacci+representations%22&sort=&language=&go=Search
>>
>>
>> I decided to try to supply these with proper definitions. These
>> sequences are found in the following paper:
>>
>> http://www.fq.math.ca/Scanned/11-4/carlitz.pdf
>>
>> Almost all of the sequences are found in some tables at the end.
>>
>> I have run into a couple of issues, though:
>>
>> 1. The function listed in the tables as lambda-prime is in OEIS as
>> A003253. However, in 2000, the sequence was marked as an erroneous
>> version of A001651. I don't understand this since the terms in A003253
>> agree with my own calculations. Can anyone shed light on this? It looks
>> to me like the sequence should be revived.
>>
>> 2. Many of the sequences listed in the table are not in OEIS, and I have
>> been trying to decide whether they should be added. The listed sequences
>> don't seem all that interesting, to me at least, and most of them are
>> just complements of each other. On the other hand, they are explicitly
>> tabulated in the paper. I don't know what the original rationale was for
>> including some but not all of the sequences. Any advice on this would be
>> appreciated.
>>
>> Here's a bunch of Sage code to compute the sequences.
>> Functions marked ### are from the paper; the rest are auxiliary.
>>
>> # Determine whether n is in range of monotonic function f
>> def isfunc(f, n) :
>>      m = 1
>>      while f(m) < n : m += 1
>>      return f(m) == n
>>
>> # Compute n-th term of complement of montonic f
>> @CachedFunction
>> def funcprime(f, n) :
>>      m = 1 if n==1 else funcprime(f,n-1)+1
>>      while isfunc(f, m) : m += 1
>>      return m
>>
>> # Compute least m such that f(m) >= targfunc(n)
>> @CachedFunction
>> def funcpre(f, targfunc, n) :
>>      m = 1 if n==1 else funcpre(f, targfunc, n-1)+1
>>      target = targfunc(n)
>>      while f(m) < target : m += 1
>>      return m
>>
>> ### A000201
>> def a(n) : return floor(golden_ratio*n)
>>
>> ### A001950
>> def b(n) : return floor(golden_ratio^2*n)
>>
>> ### A003231
>> def c(n) : return b(n) + n
>>
>> def cprime(n) : return funcprime(c, n)
>>
>> ### A003234
>> @CachedFunction
>> def s(n) :
>>      m = 1 if n==1 else s(n-1)+1
>>      while c(b(m)) != b(c(m)) - 1 : m += 1
>>      return m
>>
>> ### A003233
>> @CachedFunction
>> def r(n) :
>>      m = 1 if n==1 else r(n-1)+1
>>      while c(b(m)) != b(c(m)) : m += 1
>>      return m
>>
>> ### A003250
>> def z(n) : return ceil(1/golden_ratio^2 * c(s(n)))
>>
>> ### A003251
>> def zprime(n) : return funcprime(z, n)
>>
>> ### A003248
>> def tprime(n) : return a(s(n)) + n
>>
>> ### A003247
>> def t(n) : return funcprime(tprime, n)
>>
>> ### A003249
>> def uprime(n) : return b(s(n)) + 1
>>
>> ### not in OEIS (though used to define v, which is)
>> def u(n) : return funcprime(uprime, n)
>>
>> ### A005206
>> @CachedFunction
>> def e(n) : return 0 if n==0 else n-e(e(n-1))
>>
>> def es(n) : return e(s(n))
>>
>> ### A003254
>> def p(n) : return funcpre(r, es, n)
>>
>> ### A003255
>> def pprime(n) : return funcprime(p, n)
>>
>> ### A003256
>> def v(n) : return funcpre(u, b, n)
>>
>> ### A003257
>> def vprime(n) : return funcprime(v, n)
>>
>> def ab(n) : return a(b(n))
>> def abprime(n) : return funcprime(ab, n)
>>
>> ### not in OEIS
>> def w(n) : return funcpre(u, abprime, n)
>>
>> ### not in OEIS
>> def wprime(n) : return funcprime(w, n)
>>
>> ### not in OEIS
>> def x(n) : return funcpre(u, es, n)
>>
>> ### not in OEIS
>> def xprime(n) : return funcprime(x, n)
>>
>> def uw(n) : return u(w(n))
>>
>> ### not in OEIS
>> def y(n) : return funcpre(uw, es, n)
>>
>> ### not in OEIS
>> def yprime(n) : return funcprime(y, n)
>>
>> ### A003252
>> def lamb(n) : return funcpre(zprime, c, n)
>>
>> ### A003253
>> def lambprime(n) : return funcprime(lamb, n)
>>
>> def ec(n) : return e(c(n))
>>
>> ### A003258
>> def phi(n) : return funcpre(cprime, ec, n)
>>
>> ### A003259
>> def phiprime(n) : return funcprime(phi, n)
>>
>> ### not in OEIS
>> def psi(n) : return e(phiprime(n))
>>
>> ### not in OEIS
>> def psiprime(n) : return funcprime(psi, n)
>>
>> ### not in OEIS
>> def sigma(n) : return p(t(n))
>>
>> ### not in OEIS
>> def sigmaprime(n) : return funcprime(sigma, n)
>>
>> ### not in OEIS
>> def tau(n) : return sigma(n) - isfunc(u, n)
>>
>> ### not in OEIS
>> def tauprime(n) : return funcprime(tau, n)
>>
>> ### not in OEIS
>> def K(n) : return funcpre(b, c, n) - 1
>>
>> ### not in OEIS
>> def Kprime(n) : return funcprime(K, n)
>>
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