A075420 -id:A075420 - cp's OEIS frontend

A075421 Trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

290, 318, 719, 795, 799, 1210, 3903, 4199, 4207, 4219, 4236, 4278, 4279, 4294, 4326, 4333, 4334, 4338, 4402, 4598, 4662, 4726, 5046, 5357, 6157, 6174, 7246, 7247, 7295, 7407, 7549, 8063, 8191, 9211, 12319, 12431, 12463, 12539, 15487, 16519, 16587

Offset: 1

Klaus Brockhaus, Sep 18 2002, revised Jan 28 2004

For 318 (cf. A075153), 266718 (cf. A075466) and 270798 (cf. A075467) one can prove that the base 4 trajectory does not contain a palindrome. A proof for 290 (cf. A075299) has not been found up to now. 4398859679359 is another known candidate (obtained from a remark of David J. Seal, cf. Links) for a term whose trajectory is provably palindrome-free, but is not secured that it does not join the trajectory of some term m < n. - 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 28 for k < 20000). On the other hand, the trajectories of the terms listed above do not join the trajectory of any smaller term within at least 1000 steps.
Base-4 analog of A063048 (base 10) and A075252 (base 2); subsequence of A075420.
From A.H.M. Smeets, Mar 18 2019: (Start)
David J. Seal (see LINKS) observed a cyclic pattern (length 6) in the trajectories that can be represented by an extended right regular grammar with production rules:
S -> S_a | S_b | S_c | S_d | S_e | S_f,
S_a -> 1033202000232 T_a, T_a -> 222 T_a | 2302333113230
S_b -> 2022321332331 T_b, T_b -> 111 T_b | 1223001203131
S_c -> 10002003002212 T_c, T_c -> 222 T_c | 3221333101333
S_d -> 103312202321111 T_d, T_d -> 111 T_d | 1102023122000
S_e -> 110200123122222 T_e, T_e -> 222 T_e | 2231232001301
S_f -> 213301021321111 T_f, T_f -> 111 T_f | 1113213003312
Within the first 471 terms of this sequence we observed three trajectories with a cyclic pattern (length 6) that can be represented by a context-free grammar with production rules:
S -> S_a | S_b | S_c | S_d | S_e | S_f,
S_a -> 10 T_a 00, T_a -> 3 T_a 0 | T_a0,
S_b -> 11 T_b 01, T_b -> 0 T_b 3 | T_b0,
S_c -> 22 T_c 12, T_c -> 0 T_c 3 | T_c0,
S_d -> 10 T_d 000, T_d -> 3 T_d 0 | T_d0,
S_e -> 11 T_e 301, T_e -> 0 T_e 3 | T_e0,
S_f -> 22 T_f 312, T_f -> 0 T_f 3 | T_f0.
The terminating strings in these context-free grammars are given by:
n     2      359                371
a(n)  318    266718             270798
T_a0  33230  33230000001033230  3323001033230
T_b0  03123  03123010001103123  0312302103123
T_c0  01313  01313120002201313  0131320201313
T_d0  33323  33323000001033323  3332300103323
T_e0  03222  03222301001103222  0322201113222
T_f0  02111  02111312002202111  0211112222111
From the fact that both, right regular grammars and context-free grammars occur, we wonder if other trajectories can be represented by context-sensitive grammars as well, by which other trajectories can be proven never to end up in a palindromic string? (End)

			719 is a term since the trajectory of 719 (presumably) does not lead to an integer which occurs in the trajectory of 290 or of 318.

A.H.M. Smeets, Table of n, a(n) for n = 1..471
Klaus Brockhaus, Illustration: Distribution of terms below 2000000
David J. Seal, Results
Index entries for sequences related to Reverse and Add!

Cf. A063048, A075252, A075420, A075299, A075153, A075466, A075467, A091675.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0, 17000], (x = NestWhileList[# + IntegerReverse[#, 4] &, #, # !=IntegerReverse[#, 4] & , 1, limit];
   If[Length[x] >= limit  && Intersection[x, utraj] == {},
    utraj = Union[utraj, x]; True,
utraj = Union[utraj, x]]) &] (* Robert Price, Oct 16 2019 *)

A075153 Trajectory of 318 under the Reverse and Add! operation carried out in base 4, written in base 10.

Original entry on oeis.org

318, 1071, 5040, 5985, 10710, 20400, 24225, 43350, 81600, 85425, 165750, 327360, 342705, 664950, 1309440, 1324785, 2629110, 5241600, 5303025, 10524150, 20966400, 21027825, 41973750, 83880960, 84126705, 167925750, 335523840

Offset: 0

Klaus Brockhaus, Sep 05 2002

290 is conjectured (cf. A066450) to be the smallest number such that the Reverse and Add! algorithm in base 4 does not lead to a palindrome. 318 (not 255 since 255 is a base 4 palindrome) is up to now the smallest number whose base 4 trajectory provably does not contain a palindrome. 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 3 in {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 3 = 0.

			318 (decimal) = 10332 -> 10332 + 23301 = 100233 = 1071 (decimal).

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
K. S. Brown, Digit Reversal Sums Leading to Palindromes
Index entries for sequences related to Reverse and Add!

Cf. A058042 (trajectory of binary number 10110 (decimal 22)), A061561 (A058042 written in base 10), A066450 (conjectured minimal k so that the trajectory of k in base n does not lead to a palindrome).
Cf. A075253 (trajectory of 77 in base 2), A075420 (trajectory of n in base 4 (presumably) does not reach a palindrome), A075421 (trajectory of n in base 4 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n), A075299 (trajectory of 290 in base 4), A075466 (trajectory of 266718 in base 4), A075467 (trajectory of 270798 in base 4), A076247 (trajectory of 1059774 in base 4), A076248 (trajectory of 1059831 in base 4), A091675 (trajectory of n in base 4 (presumably) does not join the trajectory of any m < n).
Cf. A166912 (a(6*n)/3), A166913 (a(6*n+1)/3), A166914 (a(6*n+2)/240), A166915 (a(6*n+3)/15), A166916 (a(6*n+4)/30), A166917 (a(6*n+5)/240).

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(318, 26, 4);

NestWhileList[# + IntegerReverse[#, 4] &, 318,  # !=
IntegerReverse[#, 4] &, 1, 26] (* Robert Price, Oct 18 2019 *)

{m=318; stop=29; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

a(0) = 318; a(1) = 1071; for n > 1 and n = 2 (mod 6): a(n) = 5*4^(2*k+5)-5*4^(k+2) where k = (n-2)/6; n = 3 (mod 6): a(n) = 5*4^(2*k+5)+55*4^(k+2)-15 where k = (n-3)/6; n = 4 (mod 6): a(n) = 10*4^(2*k+5)+30*4^(k+2)-10 where k = (n-4)/6; n = 5 (mod 6): a(n) = 20*4^(2*k+5)-5*4^(k+2) where k = (n-5)/6; n = 0 (mod 6): a(n) = 20*4^(2*k+5)+235*4^(k+2)-15 where k = (n-6)/6; n = 1 (mod 6): a(n) = 40*4^(2*k+5)+150*4^(k+2)-10 where k = (n-7)/6.

G.f.: 3*(106 +357*x +1680*x^2 +1465*x^3 +1785*x^4 -1600*x^5 -1900*x^6 -3400*x^7 -6800*x^8 -9780*x^9 -9860*x^10 +6720*x^11 +10064*x^12 +11088*x^13) / ((1-x)*(1+x+x^2)*(1-2*x^3)*(1+2*x^3)*(1-4*x^3)).

Two comments added, g.f. edited, MAGMA program and cross-references added by Klaus Brockhaus, Oct 26 2009

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

A075685 Reverse and Add! carried out in base 4; number of steps needed to reach a palindrome, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 2, 1, 1, 0, 1, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 2, 0, 3, 2, 4, 0, 4, 3, 1, 1, 0, 1, 1, 1, 0, 4, 2, 3, 0, 2, 3, 4, 0, 4, 1, 2, 1, 0, 1, 2, 4, 0, 3, 2, 2, 0, 4, 3, 4, 0, 1, 0, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 0, 1, 3, 1, 1, 1, 2, 2, 3, 3, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2

Offset: 0

Klaus Brockhaus, Sep 24 2002

Base-4 analog of A033665 (base 10) and A066057 (base 2). For values of n such that presumably a(n) = -1 see A075420.

			26 (decimal) = 122 -> 122 + 221 = 1003 -> 1003 + 3001 = 10010 -> 10010 + 01001 = 11011 (palindrome) = 325 (decimal) requires 3 steps, so a(26) = 3.

Index entries for sequences related to Reverse and Add!

Cf. A014192, A033665, A066057, A075420.

m := 105; stop := 1000; for n := 0 to m do c := 0; k := n; v := -1; while c < stop do a := k; rev := 0; while a > 0 do rev := 4*rev + (a mod 4); a := a div 4; end; if k = rev then v := c; c := stop; else inc(c); k := k + rev; end; end; write(v," "); end;.

A075686 In base 4, 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, 4, 7, 26, 28, 127, 199, 296, 511, 3119, 16861, 18164, 19453, 20468, 270824

Offset: 0

Klaus Brockhaus, Sep 24 2002

RECORDS transform of A075685. - Base-4 analog of A065198 (base 10) and A066144 (base 2). Integers like 290, for which a palindrome is (presumably) never reached (cf. A075420), are of course disregarded. A075687 gives the corresponding records.

			Starting with 26, 3 Reverse and Add! steps are needed to reach a palindrome; starting with n < 26, less (at most 2) steps are needed.

N. J. A. Sloane, Transforms
Index entries for sequences related to Reverse and Add!

Cf. A065198, A066144, A075420, A075685, A075687.

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 = ", ")

A091680 Smallest number whose base-4 Reverse and Add! trajectory (presumably) contains exactly n base-4 palindromes, or -1 if there is no such number.

Original entry on oeis.org

290, 78, 18, 6, 3, 36, 21, 19, 7, 8, 4, 2, 1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 0

Klaus Brockhaus, Jan 28 2004

Conjecture 1: For each k > 0 the trajectory of k eventually leads to a term in the trajectory of some j which belongs to A075421, i.e., whose trajectory (presumably) never leads to a palindrome. Conjecture 2: There is no k > 0 such that the trajectory of k contains more than twelve palindromes, i.e., a(n) = -1 for n > 12.

Base-4 analog of A077594.

			a(4) = 3 since the trajectory of 3 contains the four palindromes 3, 15, 975, 64575 (3, 33, 33033, 3330333 in base 4) and at 20966400 joins the trajectory of 318 = A075421(2) and the trajectories of 1 (A035524) and 2 do not contain exactly four palindromes.

Index entries for sequences related to Reverse and Add!

Cf. A075299, A035524, A014192, A075420, A075421, A077594.

A306365 Trajectory of n under the Reverse and Add! operation carried out in base 5 (presumably) does not reach a palindrome.

Original entry on oeis.org

708, 718, 723, 731, 733, 743, 828, 838, 843, 851, 853, 863, 958, 963, 983, 1078, 1083, 1103, 1203, 1299, 1309, 1332, 1342, 1347, 1350, 1355, 1357, 1359, 1367, 1419, 1429, 1452, 1462, 1467, 1475, 1477, 1479, 1487, 1499, 1539, 1582, 1607, 1619, 1659, 1702, 1707, 1727, 1739, 1779, 1827, 1859, 1923, 1933, 1956

Offset: 1

A.H.M. Smeets, Feb 10 2019

Base-5 analog of A066059 (base 2), A077404 (base 3), A075420 (base 4) and A023108 (base 10).

All terms are tested up to 200 iteration steps, i.e., within 200 steps no palindrome was reached.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A066059, A075420, A077404, A023108.

cp's OEIS Frontend

A075421 Trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

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A075153 Trajectory of 318 under the Reverse and Add! operation carried out in base 4, written in base 10.

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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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A075685 Reverse and Add! carried out in base 4; number of steps needed to reach a palindrome, or -1 if no palindrome is ever reached.

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A075686 In base 4, 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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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.

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A091680 Smallest number whose base-4 Reverse and Add! trajectory (presumably) contains exactly n base-4 palindromes, or -1 if there is no such number.

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A306365 Trajectory of n under the Reverse and Add! operation carried out in base 5 (presumably) does not reach a palindrome.

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