# Talk:Calkin-Wilf sequence

Latest comment: 2 years ago by Paddy3118 in topic More Calculations: Python

## Off by one error?

The wikipedia entry starts the series with 1 not zero. The calculation of what term represents a rational also seems off by one. --Paddy3118 (talk) 22:29, 28 December 2020 (UTC)

Other calculations fail as 0 is never a term if calculating the i'th term from the run length encodings of i for example. Best to correct the task wording and adjust all examples I think. --Paddy3118 (talk) 23:37, 28 December 2020 (UTC)

The 123456789'th term of 83116 / 51639 applies to the wikipedia series where the "first" term is 1.

I get, using the wikipedia calcs:

```for i in [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]:
print(i, ith_term(i))
0 1
1 1
2 1/2
3 2
4 1/3
5 3/2
6 2/3
7 3
8 1/4
9 4/3```

zeroth, and first terms are both 1. Best to do as wikepedia does and have istart at 1 for the series I think? --Paddy3118 (talk) 00:07, 29 December 2020 (UTC)

When I try this, I get 'error, ith_term has not been defined'. To be slightly less churlish, I see wp defines "starting from q1 = 1" and there simply is no "zeroth" term. A q0 of 1 is just as wrong, in fact even wronger, and I'd like to see that ith_term - if it is using the formula I see on wp and in the task description it would a) be wrong and b) not be possible without assuming a q0 of 0. In my entry I cheekily went printf(1,"The first 21 terms of the Calkin-Wilf sequence are:\n 0: 0\n") just to match everyone else. Perhaps the task could be amended to say "you can quietly assume a q0 of 0 to simplify calculations but do not show it". Lastly, when you say "seems off by one" the wikipedia page clearly links 4/3 and 9 and 3/4 and 14 so... --Pete Lomax (talk) 00:37, 29 December 2020 (UTC)
I read the wp entry some more as well as others, and I agree, there is no zero'th indexed item. The series starts from the 1-indexed item which has a value of 1. Different methods of arriving at the i'th term, for i being one of all positive integers not including zero, agree. Extrapolating to a zero'th term do not, and have no meaning in terms of the tree that is traversed to form the series.
I could amend the task description... --Paddy3118 (talk) 05:24, 29 December 2020 (UTC)
Description fixed. Examples flagged. It is a great task. This is just a minor change. --Paddy3118 (talk) 05:55, 29 December 2020 (UTC)
That all seems fine. Cheers. Thebigh (talk) 08:00, 29 December 2020 (UTC)

## More Calculations: Python

I enjoyed learning about the sequence; especially the use of run-length encoded binaries and continued fractions from the Wikipedia page. I coded four ways of generating the sequence as well as the full method of finding the index to any rational in the sequence.
I don't think it's enough to create a separate task from, so I park it here:

<lang python>from fractions import Fraction from math import floor from itertools import islice, groupby from typing import List from random import randint

def cw_floor() -> Fraction:

```   "Calkin-Wilf sequence generator (uses floor function)"
a = Fraction(1)
while True:
yield a
a = 1 / (2 * floor(a) + 1 - a)
```

def cw_mod() -> Fraction:

```   """\
Calkin-Wilf sequence generator (uses modulo function)
See: https://math.stackexchange.com/a/3298088/55677"""
a, b = 1, 1
while True:
yield Fraction(a, b)
a, b = b, a - 2*(a%b) + b
```

def cw_direct(i: int) -> Fraction:

```   "Calkin-Wilf sequence generation directly from index"
as_bin = f"{i:b}"
run_len_encoded = [len(list(g))
for k,g in groupby(reversed(as_bin))]
if as_bin[-1] == '0':   # Correction for even i by inserting zero 1's
run_len_encoded.insert(0, 0)
return _continued_frac_to_fraction((run_len_encoded))

```

def _continued_frac_to_fraction(cf):

```   ans = Fraction(cf[-1])
for term in reversed(cf[:-1]):
ans = term + 1 / ans
return ans
```

def get_cw_terms_index(f: Fraction) -> int:

```   "Given f return the index of where it occurs in the Calkin-Wilf sequence"
ans, dig, pwr = 0, 1, 0
for n in _frac_to_odd_continued_frac(f):
for _ in range(n):
ans |= dig << pwr
pwr += 1
dig ^= 1
return ans
```

def _frac_to_odd_continued_frac(f: Fraction) -> List[int]:

```   num, den = f.as_integer_ratio()
ans = []
while den:
num, (digit, den) = den, divmod(num, den)
ans.append(digit)
if len(ans) %2 == 0:    # Must be odd length
ans[-1:] = [ans[-1] -1, 1]
return ans
```

def fusc() -> List[int]:

```   "Fusc sequence generator."
f = 
yield f
f.append(1)
yield f
n = 2
while True:
fn2 = f[n // 2]
f.append(fn2)
yield f
f.append(fn2 + f[n // 2 + 1])
yield f
n += 2
```

def cw_fusc() -> Fraction:

```   "Calkin-Wilf sequence generator (uses fusc generator)"
f = fusc()
next(f); next(f)
for series in f:
yield Fraction(*series[-2:])
```

if __name__ == '__main__':

```   n = 10_000
print(f"Checking {n:_} terms calculated in four ways:")
using_floor  = list(islice(cw_floor(), n))
using_mod    = list(islice(cw_mod(), n))
using_direct = [cw_direct(i) for i in range(1, n+1)]
using_fusc   = list(islice(cw_fusc(), n))
if using_floor == using_mod == using_direct == using_fusc:
print('  OK.')
print('  FIRST 15 TERMS:', ', '.join(str(x) for x in using_direct[:15]))
# Indices of successive terms
print('  CHECKING SUCCESSIVE TERMS ARE FROM SUCCESSIVE INDICES: ')
first_index = randint(999, 999_999_999)
#terms = [Fraction(83116, 51639), Fraction(51639, 71801)]
terms = [cw_direct(first_index), cw_direct(first_index + 1)]
indices = [get_cw_terms_index(t) for t in terms]
nth_terms = [cw_direct(index) for index in indices]
if terms == nth_terms and indices + 1 == indices:
for t, i in zip(terms, indices):
print(f"    {t} is the {i:_}'th term.")
else:
print('    Whoops! Problems in finding indices of '
"successive terms.")
else:
print('Whoops! Calculation methods do not match.')</lang>
```
Output:
```Checking 10_000 terms calculated in four ways:
OK.
FIRST 15 TERMS: 1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4
CHECKING SUCCESSIVE TERMS ARE FROM SUCCESSIVE INDICES:
13969/9194 is the 416_907_269'th term.
9194/13613 is the 416_907_270'th term.```