A066059 -id:A066059 - cp's OEIS frontend

A061561 Trajectory of 22 under the Reverse and Add! operation carried out in base 2.

22, 35, 84, 105, 180, 225, 360, 405, 744, 837, 1488, 1581, 3024, 3213, 6048, 6237, 12192, 12573, 24384, 24765, 48960, 49725, 97920, 98685, 196224, 197757, 392448, 393981, 785664, 788733, 1571328, 1574397, 3144192, 3150333, 6288384, 6294525

Offset: 0

N. J. A. Sloane, May 18 2001

Sequence A058042 written in base 10. 22 is the smallest number whose base 2 trajectory does not contain a palindrome.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 0.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 1. - Klaus Brockhaus, Dec 09 2009

T. D. Noe, Table of n, a(n) for n=0..500
Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!

Cf. A035522 (trajectory of 1 in base 2), A058042 (trajectory of 22 in base 2, written in base 2), A075253 (trajectory of 77 in base 2), A075268 (trajectory of 442 in base 2), A077076 (trajectory of 537 in base 2), A077077 (trajectory of 775 in base 2), A066059 (trajectory of n in base 2 (presumably) does not reach a palindrome), A075252 (trajectory of n in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n), A075153 (trajectory of 318 in base 4).

Cf. A171470 (a(4*n)/2), A171471 (a(4*n+1)), A171472 (a(4*n+2)/12), A171473 (a(4*n+3)/3).

m := 22; stop := 36; c := 0; while c < stop do write(m,","); k := bit_length(m); rev := 0; for i := 0 to k-1 do if bit_test(m,i) then rev := bit_set(rev,k-1-i); end; end; inc(c); m := m+rev; end;.

a061561 n = a061561_list !! n
a061561_list = iterate a055944 22  -- Reinhard Zumkeller, Apr 21 2013

trajectory:=function(init, steps, base) a:=init; S:=[a]; for n in [1..steps] do a+:=Seqint(Reverse(Intseq(a,base)),base); Append(~S, a); end for; return S; end function; trajectory(22, 35, 2); // Klaus Brockhaus, Dec 09 2009

binRA[n_] := If[Reverse[IntegerDigits[n, 2]] == IntegerDigits[n, 2], n, FromDigits[Reverse[IntegerDigits[n, 2]], 2] + n]; NestList[binRA, 22, 100] (* Adapted from Ben Branman's code for A213012, Alonso del Arte, Jun 02 2012 *)

{m=22; stop=36; c=0; while(c0,d=divrem(k,2); k=d[1]; rev=2*rev+d[2]); c++; m=m+rev)}

a(0) = 22; a(1) = 35; for n > 1 and n = 2 (mod 4): a(n) = 6*2^(2*k)-3*2^k where k = (n+6)/4; n = 3 (mod 4): a(n) = 6*2^(2*k)+3*2^k-3 where k = (n+5)/4; n = 0 (mod 4): a(n) = 12*2^(2*k)-3*2^k where k = (n+4)/4; n = 1 (mod 4): a(n) = 12*2^(2*k)+9*2^k-3 where k = (n+3)/4. [Klaus Brockhaus, Sep 05 2002]
G.f.: (22+35*x+18*x^2-72*x^4-90*x^5-48*x^6-60*x^7+80*x^8+112*x^9) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)). [Klaus Brockhaus, Sep 05 2002, edited Dec 09 2009]
a(n+1) = A055944(a(n)). - Reinhard Zumkeller, Apr 21 2013

More terms from Klaus Brockhaus, May 27 2001

A075252 Trajectory of n under the Reverse and Add! operation carried out in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

Original entry on oeis.org

22, 77, 442, 537, 775, 1066, 1081, 1082, 1085, 1115, 1562, 1575, 1587, 2173, 3355, 3599, 3871, 4099, 4153, 4185, 4193, 4202, 4262, 4285, 4402, 4633, 4666, 6163, 6166, 6374, 9241, 9466, 16544, 16546, 16586, 16601, 16613, 16616, 16720, 16748, 16994

Offset: 1

Klaus Brockhaus, Sep 10 2002

Base-2 analog of A063048 (base 10) and A075421 (base 4); subsequence of A066059. - For the trajectory of 22 (cf. A061561) and the trajectory of 77 (cf. A075253) it has been proved that they do not contain a palindrome. A similar proof can be given for most terms of this sequence, but there are a few terms (4262, 17498, 33378, 33898, ...) whose trajectory does not show the kind of regularity that can be utilized for the construction of a proof. - If the trajectory of an integer k joins the trajectory of a smaller integer which is a term of the present sequence, then this occurs after very few 'Reverse and Add!' steps (at most 84 for numbers < 20000). On the other hand, the trajectories of the terms of this sequence do not join the trajectory of any smaller term within at least 1000 steps.
From A.H.M. Smeets, Feb 12 2019: (Start)
Most terms in this sequence eventually give rise to a regular binary pattern. These regular patterns can be represented by contextfree grammars:
S_a -> 10 T_a 00, T_a -> 1 T_a 0 | A_a(n);
S_b -> 11 T_b 01, T_b -> 0 T_b 1 | B_a(n);
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | C_a(n) and
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | D_a(n).
A_22 = 1101, B_22 = 1000, C_22 = 1101, D_22 = 0010 (see also A058042);
A_77 = 1100010, B_77 = 0000101, C_77 = 1101011, D_77 = 0100000 (see also A075253)
Decimal representations for 10 A_a(n) 00 are given by A306514(n).
Binary representations for 10 A_a(n) 00 are given by A306515(n).
(End)

			442 is a term since the trajectory of 442 (presumably) does not lead to an integer which occurs in the trajectory of 22 or of 77.

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!

Cf. A063048, A075421, A066059, A058042, A061561, A075253, A306514, A306515.

limit = 10^2; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0, 17000], (x = NestWhileList[# + IntegerReverse[#, 2] &, #, # != IntegerReverse[#, 2] & , 1, limit];
   If[Length[x] >= limit  && Intersection[x, utraj] == {},
    utraj = Union[utraj, x]; True,
utraj = Union[utraj, x]]) &] (* Robert Price, Oct 16 2019 *)

A075253 Trajectory of 77 under the Reverse and Add! operation carried out in base 2.

Original entry on oeis.org

77, 166, 267, 684, 897, 1416, 1557, 2904, 3333, 5904, 6189, 11952, 12813, 24096, 24669, 48480, 50205, 97344, 98493, 195264, 198717, 391296, 393597, 783744, 790653, 1569024, 1573629, 3140352, 3154173, 6283776, 6292989, 12572160

Offset: 0

Klaus Brockhaus, Sep 10 2002

22 is the smallest number whose base 2 trajectory (A061561) provably does not contain a palindrome. 77 is the next number (cf. A075252) with a completely different trajectory which has this property. A proof along the lines of Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2, can be based on the formula given below.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 1.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 0.
Interleaving of A176632, 2*A176633, 3*A176634, 12*A176635.
From A.H.M. Smeets, Feb 11 2019: (Start)
Pattern with cycle length 4 in binary representation, represented by contextfree grammars with production rules:
S_a -> 10 T_a 00, T_a -> 1 T_a 0 | 1100010;
S_b -> 11 T_b 01, T_b -> 0 T_b 1 | 0000101;
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | 1101011;
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | 0100000;
the trajectory is similar to that of 22 (see A058042) except for the stopping strings in T_a, T_b, T_c and T_d. (End)

			267 (decimal) = 100001011 -> 100001011 + 110100001 = 1010101100 = 684 (decimal).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!
Index entries for linear recurrences with constant coefficients, signature (0, 3, 0, 0, 0, -6, 0, 4).

Cf. A061561 (trajectory of 22 in base 2), A075268 (trajectory of 442 in base 2), A077076 (trajectory of 537 in base 2), A077077 (trajectory of 775 in base 2), A066059 (trajectory of n in base 2 presumably does not reach a palindrome), A075252 (trajectory of n in base 2 does not reach a palindrome and presumably does not join the trajectory of any term m < n), A092210 (trajectory of n in base 2 presumably does not join the trajectory of any m < n).

Cf. A176632 (a(4*n)), A176633 (a(4*n+1)/2), A176634 (a(4*n+2)/3), A176635 (a(4*n+3)/12).

a075253 n = a075253_list !! n
a075253_list = iterate a055944 77  -- Reinhard Zumkeller, Apr 21 2013

trajectory:=function(init, steps, base) S:=[init]; a:=S[1]; for n in [1..steps] do a+:=Seqint(Reverse(Intseq(a,base)),base); Append(~S, a); end for; return S; end function; trajectory(77, 31, 2);

seq(coeff(series((77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6-348*x^7-44*x^8+632*x^9+504*x^10)/((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Feb 12 2019

CoefficientList[Series[(77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6 -348*x^7-44*x^8 +632*x^9+504*x^10)/((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)), {x,0,40}], x] (* G. C. Greubel, Feb 11 2019 *)
NestWhileList[# + IntegerReverse[#, 2] &, 77,  # !=
IntegerReverse[#, 2] &, 1, 31] (* Robert Price, Oct 18 2019 *)

{m=77; stop=34; c=0; while(c0,d=divrem(k,2); k=d[1]; rev=2*rev+d[2]); c++; m=m+rev)}

((77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6 -348*x^7-44*x^8 +632*x^9+504*x^10)/((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 11 2019

a(0) = 77; a(1) = 166; a(2) = 267; for n > 2 and
n = 3 (mod 4): a(n) = 48*2^(2*k)-21*2^k where k = (n+5)/4;
n = 0 (mod 4): a(n) = 48*2^(2*k)+33*2^k-3 where k = (n+4)/4;
n = 1 (mod 4): a(n) = 96*2^(2*k)-30*2^k where k = (n+3)/4;
n = 2 (mod 4): a(n) = 96*2^(2*k)+6*2^k-3 where k = (n+2)/4.
G.f.: (77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6-348*x^7-44*x^8 +632*x^9+504*x^10) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
G.f. for the sequence starting at a(3): 3*x^3*(228+299*x-212*x^2 -378*x^3-448*x^4-446*x^5+432*x^6+524*x^7) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
a(n+1) = A055944(a(n)). - Reinhard Zumkeller, Apr 21 2013

Three comments added, g.f. edited, MAGMA program and crossrefs added by Klaus Brockhaus, Apr 25 2010

A075420 Trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not reach a palindrome.

Original entry on oeis.org

290, 318, 378, 381, 438, 444, 462, 498, 501, 504, 510, 545, 567, 573, 627, 633, 636, 639, 693, 696, 699, 717, 719, 732, 751, 753, 756, 759, 765, 775, 795, 799, 800, 822, 823, 828, 835, 847, 859, 882, 883, 888, 894, 895, 915, 919, 927, 948, 954, 967, 972

Offset: 1

Klaus Brockhaus, Sep 18 2002

Base-4 analog of A023108 (base 10) and A066059 (base 2).

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
Index entries for sequences related to Reverse and Add!

Cf. A023108, A066059, A075421, A075299, A075153.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
Select[Range[1000],
Length@NestWhileList[# + IntegerReverse[#, 4] &, #, # !=
IntegerReverse[#, 4]  &, 1, limit] == limit + 1 &] (* Robert Price, Oct 16 2019 *)

{stop=1000; for(n=1,980,k=n; c=0; while(c0,d=divrem(a,4); a=d[1]; rev=4*rev+d[2]); if(rev==k,c=stop+1,k=k+rev; c++)); if(c==stop,print1(n,",")))}

Offset changed to 1 by A.H.M. Smeets, Feb 10 2019

A077404 Trajectory of n under the Reverse and Add! operation carried out in base 3 (presumably) does not reach a palindrome.

Original entry on oeis.org

103, 127, 167, 176, 180, 183, 185, 191, 197, 200, 206, 207, 209, 215, 221, 230, 239, 291, 309, 318, 340, 358, 369, 376, 387, 394, 396, 413, 418, 431, 436, 449, 454, 467, 472, 582, 600, 618, 636, 655, 660, 673, 678, 691, 696, 709, 714, 746, 776, 803, 805

Offset: 1

Klaus Brockhaus, Nov 05 2002

Base-3 analog of A066059 (base 2), A075420 (base 4) and A023108 (base 10).

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
Index entries for sequences related to Reverse and Add!

Cf. A023108, A066059, A075420, A077405.

m := 900; stop := 1000; for n := 0 to m do c := 0; k := n; b := 1; while b do d := k; rev := 0; while d > 0 do rev := 3*rev + (d mod 3); d := d div 3; end; if k = rev or c = stop then b := 0; else k := k + rev; end; inc(c); end; if c > stop then write(n,","); end; end;

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
Select[Range[805],
 Length@NestWhileList[# + IntegerReverse[#, 3] &, #, # !=
IntegerReverse[#, 3]  &, 1, limit] == limit + 1 &] (* Robert Price, Oct 19 2019 *)

Offset changed to 1 by A.H.M. Smeets, Feb 10 2019

A066144 In base 2: n sets a new record for the number of 'Reverse and Add' steps needed to reach a palindrome starting with n.

Original entry on oeis.org

0, 2, 11, 19, 20, 74, 398, 779, 1062, 2329, 4189, 4280, 11278, 19962, 98318, 135137, 1051360, 1592930, 69226926, 4295054186, 4446008678, 17187271449, 18123849698

Offset: 1

Klaus Brockhaus, Dec 08 2001

The analog of A065198 in base 2. Integers like 22, for which a palindrome is never reached (cf. A066059), are of course disregarded. A066145 gives the corresponding records.

			Starting with 74, 11 'Reverse and Add' steps are needed to reach a palindrome; starting with n < 74, less (at most 5) steps are needed.

Index entries for sequences related to Reverse and Add!

Record setting values in base b: A077406 (b=3), A075686 (b=4), A306599 (b=8), A065198 (b=10), A348571 (Zeckendorf).

Cf. A062130, A066057, A066058, A066059, A066145.

limit = 10^4; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
best = -1; Select[Range[0, 100000], (np = #; i = 0;
   While[np != IntegerReverse[np, 2] && i < limit,
    np = np + IntegerReverse[np, 2]; i++];
If[i >= limit, False, If[i > best, best = i; True]]) &] (* Robert Price, Oct 14 2019 *)

Offset corrected and a(19)-a(23) from A.H.M. Smeets, Apr 30 2022

A066145 In base 2, records for the number of 'Reverse and Add' steps needed to reach a palindrome.

Original entry on oeis.org

0, 1, 2, 4, 5, 11, 21, 32, 37, 46, 48, 49, 53, 89, 99, 142, 147, 273, 297, 345, 515, 550, 573

Offset: 1

Klaus Brockhaus, Dec 08 2001

The analog of A065199 in base 2. A066144 gives the corresponding starting points.

Terms a(19..22) obtained by assuming that a(n+1) <= a(n) + 300. - A.H.M. Smeets, Apr 30 2022

			Starting with 74, 11 'Reverse and Add' steps are needed to reach a palindrome; starting with n < 74, at most 5 steps are needed.

Index entries for sequences related to Reverse and Add!

Record values in base b: A077407 (b=3), A075687 (b=4), A306600 (b=8), A065199 (b=10), A348572 (Zeckendorf).

Cf. A062130, A066057, A066058, A066059, A066144.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
best = -1; lst = {};
For[n = 0, n <= 10000, n++,
np = n; i = 0;
While[np != IntegerReverse[np, 2] && i < limit,
  np = np + IntegerReverse[np, 2]; i++];
If[i < limit && i > best, best = i; AppendTo[lst, i]]]; lst (* Robert Price, Oct 14 2019 *)

Offset corrected and a(19)-a(23) by A.H.M. Smeets, Apr 30 2022

A306514 Decimal representation of binary numbers with string structure 10s00, s in {0,1}*, such that it results in a non-palindromic cycle of length 4 in the Reverse and Add! procedure in base 2.

Original entry on oeis.org

84, 180, 324, 360, 684, 744, 1416, 1488, 2628, 2904, 3024, 5580, 5904, 6048, 10836, 11400, 11952, 12192, 21060, 21684, 23220, 23448, 23556, 24096, 24384, 43188, 43668, 44604, 44748, 46248, 46260, 47376, 48480, 48960, 86388, 86964, 91272, 92520, 92532, 93108, 95592, 96048, 96264, 97344, 97920, 166212

Offset: 1

A.H.M. Smeets, Feb 21 2019

If the decimal representation of the binary string 10s00 is in the sequence, so is 101s000.
For binary representation see A306515.
This sequence is a subset of A066059.
These regular patterns can be represented by the context-free grammar with production rules:
S -> S_a | S_b | S_c | S_d
S_a -> 10 T_a 00,  T_a -> 1 T_a 0 | T_a0,
S_b -> 11 T_b 01,  T_b -> 0 T_b 1 | T_b0,
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | T_c0,
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | T_d0,
where T_a0, T_b0, T_c0 and T_d0 are some terminating strings.
Numbers in this sequence are generated by choosing S_a or S_c from the starting symbol S.
The decimal representation of all binary numbers derived by S -> S_a -> 10 T_a 00 -> 10 T_a0 00 are given in sequence A306516, its binary representation in A306517.
Observed: all values are in the ranges lower(k)..upper(k), where lower(k) = 81*2^k + 2^floor((k+6)/2) + 2^6*(2^floor((k-8)/2) - 1) + 4, which holds for k >= 11, and upper(k) = 3*2^floor((k+4)/2)*(2^floor((k+7)/2) - 1), which holds for k >= 0; the number of terms in each successive range increases by about a factor of 4/3. All terms between lower(k) and upper(k) are represented by a (k+7)-binary-digit number (see A306515). Each m-binary-digit number will have a successive number of m+1 binary digits in the next range. About 1/4 of each obtained number in this sequence has a new unique cyclic trajectory (see A306516 and A306517), i.e., a cyclic trajectory not joining a previous cyclic trajectory, which explains the growth factor of 4/3 for each successive range.
All terms A061561(4k+2) for k >= 0 are included in this sequence.
All values in A103897(k+3) for k >= 0 are included in this sequence.

			a(45) = 97920 = upper(10)
The following burst of terms is from a(46) = 166212 = lower(11) up to and including a(60) = 196224 = upper(12).
The burst of terms corresponding with k = 28 is from lower(28) = 21743468484 = a(5276) up to and including upper(28) = 25769607168 = a(6976).

A.H.M. Smeets, Table of n, a(n) for n = 1..6976

Cf. A061561, A103897, A306515, A306516, A306517.

a(n) = 0 (mod 12).

A306515 Binary numbers with string structure 10s00, s in {0,1}*, such that it results in a non-palindromic cycle of length 4 in the Reverse and Add! procedure in base 2.

Original entry on oeis.org

1010100, 10110100, 101000100, 101101000, 1010101100, 1011101000, 10110001000, 10111010000, 101001000100, 101101011000, 101111010000, 1010111001100, 1011100010000, 1011110100000, 10101001010100, 10110010001000, 10111010110000, 10111110100000, 101001001000100, 101010010110100

Offset: 1

A.H.M. Smeets, Feb 21 2019

If the binary representation of the binary string 10s00 is in the sequence, so is 101s000.
For decimal representation see A306514.
This sequence is a subset of A066059.
These regular patterns can be represented by the context-free grammar with production rules:
S -> S_a | S_b | S_c | S_d
S_a -> 10 T_a 00,  T_a -> 1 T_a 0 | T_a0,
S_b -> 11 T_b 01,  T_b -> 0 T_b 1 | T_b0,
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | T_c0,
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | T_d0,
where T_a0, T_b0, T_c0 and T_d0 are some terminating strings.
Numbers in this sequence are generated by choosing S_a or S_c from the starting symbol S.
From the fact that all strings derived from S_b have prefix 11 and suffix 00, it can be proved that all strings derived from S_a must have prefix 111 (i.e., 1 is prefix of s, with s as in the name of this sequence). Similarly, from the fact that all strings derived from S_d have prefix 11 and suffix 000, it can be proved that all strings derived from S_c must have prefix 111 (i.e., again, 1 is prefix of s, with s as in the name of this sequence). In the later case, 11 is a prefix of s, which is even stronger. I believe additional stronger conditions can be observed and proved, so I challenge others to take a look at it too.

A.H.M. Smeets, Table of n, a(n) for n = 1..6976

Cf. A306514, A306516, A306517.

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.

Original entry on oeis.org

59, 61, 69, 75, 77, 100, 105, 113, 115, 122, 128, 130, 131, 135, 136, 140, 142, 143, 148, 151, 153, 160, 162, 163, 166, 172, 177, 180, 183, 188, 191, 192, 196, 198, 200, 209, 210, 212, 215, 222, 223, 229, 230, 231, 237, 240, 249, 250, 257, 258, 263, 264, 266

Offset: 1

A.H.M. Smeets, Oct 23 2021

Zeckendorf representation version of A023108 (base 10).
For the Zeckendorf representation of numbers see A014417.
For palindromic numbers in Zeckendorf representation see A094202.
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).
For records and record-setting values in the number of "Reverse and Add!" steps see A348572 and A348571 respectively.
Do any of these numbers have a trajectory in which the Lychrel property can be proved (like 22 in base 2 as in A061561)?
Iteration steps are given by n := n+A349238(n), or n := A349239(n).
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).
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.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
C. Ahlbach, J. Usatine, C. Frougny, and N. Pippenger, Efficient algorithms for Zeckendorf arithmetic, Fibonacci Quart. 51, no. 3 (2013), 249-255.

Cf. A014417, A061561, A094202, A348571, A348572, A349238, A349239.

Lychrel numbers in fixed bases: A066059 (base 2), A077404 (base 3), A075420 (base 4), A023108 (base 10).

# Using functions NumToFib and RevFibToNum from A349238.
n, a = 0, 0
while n < 53:
    a += 1
    aa, sa = a, NumToFib(a)
    ar, s = RevFibToNum(sa), 0
    while aa != ar and s < 10000:
        s, aa = s+1, aa+ar
        sa = NumToFib(aa)
        ar = RevFibToNum(sa)
    if aa != ar:
        n += 1
        print(a, end = ", ")

cp's OEIS Frontend

A061561 Trajectory of 22 under the Reverse and Add! operation carried out in base 2.

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A075252 Trajectory of n under the Reverse and Add! operation carried out in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

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A075253 Trajectory of 77 under the Reverse and Add! operation carried out in base 2.

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A075420 Trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not reach a palindrome.

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A077404 Trajectory of n under the Reverse and Add! operation carried out in base 3 (presumably) does not reach a palindrome.

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A066144 In base 2: n sets a new record for the number of 'Reverse and Add' steps needed to reach a palindrome starting with n.

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A066145 In base 2, records for the number of 'Reverse and Add' steps needed to reach a palindrome.

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