[seqfan] Re: Sequences related to Fibonacci representations

[Eric Schmidt]

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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