cp's OEIS Frontend

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

%I A349238 #30 Apr 01 2024 11:53:34
%S A349238 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,
%T A349238 4,25,17,12,33,1,35,22,14,48,9,43,30,6,40,27,19,53,4,38,25,17,51,12,
%U A349238 46,33,1,56,35,22,77,14,69,48,9,64,43,30,85,6,61,40,27
%N A349238 Reverse the digits in the Zeckendorf representation of n (A189920).
%C A349238 Fixed points a(n) = n are the Zeckendorf palindromes n = A094202.
%C A349238 Apart from a(0)=0, all terms end with a 1 digit so are "odd" A003622.
%C A349238 a(n) = 1 iff n is a Fibonacci number >= 1 (A000045) since they are Zeckendorf 100..00 which reverses to 00..001.
%C A349238 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.
%C A349238 The equivalent reversal in binary is A030101 so that a conversion to Fibbinary (A003714) and back gives a(n) = A022290(A030101(A003714(n))).
%C A349238 A reverse and reverse again loses any least significant 0 digits as in A348853 so that a(a(n)) = A348853(n).
%H A349238 Kevin Ryde, <a href="/A349238/b349238.txt">Table of n, a(n) for n = 0..10000</a>
%H A349238 Kevin Ryde, <a href="/A349238/a349238.gp.txt">PARI/GP Code</a>.
%F A349238 There is a linear representation of rank 6 for this sequence. - _Jeffrey Shallit_, May 13 2023
%e A349238 n    = 1445 = Zeckendorf 101000101001000
%e A349238 a(n) =  313 = Zeckendorf 000100101000101 reversal
%o A349238 (PARI) \\ See links.
%o A349238 (Python)
%o A349238 def NumToFib(n): # n > 0
%o A349238     f0, f1, k = 1, 1, 0
%o A349238     while f0 <= n:
%o A349238         f0, f1, k = f0+f1, f0, k+1
%o A349238     s = ""
%o A349238     while k > 0:
%o A349238         f0, f1, k = f1, f0-f1, k-1
%o A349238         if f0 <= n:
%o A349238             s, n = s+"1", n-f0
%o A349238         else:
%o A349238             s = s+"0"
%o A349238     return s
%o A349238 def RevFibToNum(s):
%o A349238     f0, f1 = 1, 1
%o A349238     n, k = 0, 0
%o A349238     while k < len(s):
%o A349238         if s[k] == "1":
%o A349238             n = n+f0
%o A349238         f0, f1, k = f0+f1, f0, k+1
%o A349238     return n
%o A349238 n, a = 0, 0
%o A349238 print(a, end = ", ")
%o A349238 while n < 71:
%o A349238     n += 1
%o A349238     print(RevFibToNum(NumToFib(n)), end = ", ") # _A.H.M. Smeets_, Nov 14 2021
%Y A349238 Cf. A189920 (Zeckendorf digits), A094202 (fixed points), A003622 (range), A348853 (delete trailing 0's).
%Y A349238 Cf. A003714 (Fibbinary), A022290 (its inverse).
%Y A349238 Cf. A343150 (reverse below MSB).
%Y A349238 Other base reversals: A030101 (binary), A004086 (decimal).
%K A349238 base,easy,nonn
%O A349238 0,5
%A A349238 _Kevin Ryde_, Nov 11 2021