A349238 Reverse the digits in the Zeckendorf representation of n (A189920).

0, 1, 1, 1, 4, 1, 6, 4, 1, 9, 6, 4, 12, 1, 14, 9, 6, 19, 4, 17, 12, 1, 22, 14, 9, 30, 6, 27, 19, 4, 25, 17, 12, 33, 1, 35, 22, 14, 48, 9, 43, 30, 6, 40, 27, 19, 53, 4, 38, 25, 17, 51, 12, 46, 33, 1, 56, 35, 22, 77, 14, 69, 48, 9, 64, 43, 30, 85, 6, 61, 40, 27

Offset: 0

Author

Kevin Ryde, Nov 11 2021

Keywords

• base
• easy
• nonn

Comments

Fixed points a(n) = n are the Zeckendorf palindromes n = A094202.

Apart from a(0)=0, all terms end with a 1 digit so are "odd" A003622.

a(n) = 1 iff n is a Fibonacci number >= 1 (A000045) since they are Zeckendorf 100..00 which reverses to 00..001.

A given k first occurs as a(n) = k at its reversal n = a(k), and thereafter at this n with any number of least significant 0's appended.

The equivalent reversal in binary is A030101 so that a conversion to Fibbinary (A003714) and back gives a(n) = A022290(A030101(A003714(n))).

A reverse and reverse again loses any least significant 0 digits as in A348853 so that a(a(n)) = A348853(n).

Examples

			n    = 1445 = Zeckendorf 101000101001000
a(n) =  313 = Zeckendorf 000100101000101 reversal
		

Links

• Kevin Ryde, Table of n, a(n) for n = 0..10000
• Kevin Ryde, PARI/GP Code.

Crossrefs

Cf. A189920 (Zeckendorf digits), A094202 (fixed points), A003622 (range), A348853 (delete trailing 0's).

Cf. A003714 (Fibbinary), A022290 (its inverse).

Cf. A343150 (reverse below MSB).

Other base reversals: A030101 (binary), A004086 (decimal).

Programs

PARI

\\ See links.

Python

def NumToFib(n): # n > 0
    f0, f1, k = 1, 1, 0
    while f0 <= n:
        f0, f1, k = f0+f1, f0, k+1
    s = ""
    while k > 0:
        f0, f1, k = f1, f0-f1, k-1
        if f0 <= n:
            s, n = s+"1", n-f0
        else:
            s = s+"0"
    return s
def RevFibToNum(s):
    f0, f1 = 1, 1
    n, k = 0, 0
    while k < len(s):
        if s[k] == "1":
            n = n+f0
        f0, f1, k = f0+f1, f0, k+1
    return n
n, a = 0, 0
print(a, end = ", ")
while n < 71:
    n += 1
    print(RevFibToNum(NumToFib(n)), end = ", ") # A.H.M. Smeets, Nov 14 2021

Formula

There is a linear representation of rank 6 for this sequence. - Jeffrey Shallit, May 13 2023