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

0,5

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).

Links

Kevin Ryde, Table of n, a(n) for n = 0..10000

Kevin Ryde, PARI/GP Code.

Formula

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

Example

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

Prog

(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

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).

Sequence in context: A050307 A248416 A348853 * A249074 A021710 A127555

Adjacent sequences: A349235 A349236 A349237 * A349239 A349240 A349241

Keyword

base,easy,nonn

Author

Kevin Ryde, Nov 11 2021

Status

approved