[seqfan] Sequences related to Fibonacci representations

[Eric Schmidt]

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)