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A348570 Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits in Zeckendorf representation reversed). Zeckendorf representation analog of Lychrel numbers.

%I A348570 #46 Jan 05 2025 19:51:42
%S A348570 59,61,69,75,77,100,105,113,115,122,128,130,131,135,136,140,142,143,
%T A348570 148,151,153,160,162,163,166,172,177,180,183,188,191,192,196,198,200,
%U A348570 209,210,212,215,222,223,229,230,231,237,240,249,250,257,258,263,264,266
%N A348570 Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits in Zeckendorf representation reversed). Zeckendorf representation analog of Lychrel numbers.
%C A348570 Zeckendorf representation version of A023108 (base 10).
%C A348570 For the Zeckendorf representation of numbers see A014417.
%C A348570 For palindromic numbers in Zeckendorf representation see A094202.
%C A348570 The "Reverse and Add!" operation (A349239) applied in Zeckendorf representation seems to behave similarly to the "Reverse and Add!" operation applied in any fixed-base representation. The first 53 terms are however obtained after performing 10^4 "Reverse and Add!" steps (see Python program).
%C A348570 For records and record-setting values in the number of "Reverse and Add!" steps see A348572 and A348571 respectively.
%C A348570 Do any of these numbers have a trajectory in which the Lychrel property can be proved (like 22 in base 2 as in A061561)?
%C A348570 Iteration steps are given by n := n+A349238(n), or n := A349239(n).
%C A348570 Closure of reverse operation is given by: Let Z be the regular expression for numbers in Zeckendorf representation, Z = 0|(100*)*10*, and L(Z) its corresponding regular language. Then for s in L(Z), the reversal of s is in L(0*)L(Z).
%C A348570 Let h be the homomorphism from Zeckendorf representation to a conventional radix representation, then addition in Zeckendorf representation, +_Z, is given by z1 +_Z z2 = h^(-1)(h(z1) + h(z2)). A direct method for addition in Zeckendorf representation is given by Ahlbach et al.
%H A348570 A.H.M. Smeets, <a href="/A348570/b348570.txt">Table of n, a(n) for n = 1..20000</a>
%H A348570 C. Ahlbach, J. Usatine, C. Frougny, and N. Pippenger, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/51-3/AhlbachUsatineFrougnyPippenger.pdf">Efficient algorithms for Zeckendorf arithmetic</a>, Fibonacci Quart. 51, no. 3 (2013), 249-255.
%o A348570 (Python) # Using functions NumToFib and RevFibToNum from A349238.
%o A348570 n, a = 0, 0
%o A348570 while n < 53:
%o A348570     a += 1
%o A348570     aa, sa = a, NumToFib(a)
%o A348570     ar, s = RevFibToNum(sa), 0
%o A348570     while aa != ar and s < 10000:
%o A348570         s, aa = s+1, aa+ar
%o A348570         sa = NumToFib(aa)
%o A348570         ar = RevFibToNum(sa)
%o A348570     if aa != ar:
%o A348570         n += 1
%o A348570         print(a, end = ", ")
%Y A348570 Cf. A014417, A061561, A094202, A348571, A348572, A349238, A349239.
%Y A348570 Lychrel numbers in fixed bases: A066059 (base 2), A077404 (base 3), A075420 (base 4), A023108 (base 10).
%K A348570 nonn,base
%O A348570 1,1
%A A348570 _A.H.M. Smeets_, Oct 23 2021