A016016 -id:A016016 - cp's OEIS frontend

A006960 Reverse and Add! sequence starting with 196.

196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, 60744805, 111589511, 227574622, 454050344, 897100798, 1794102596, 8746117567, 16403234045, 70446464506, 130992928913, 450822227944, 900544455998, 1800098901007, 8801197801088, 17602285712176

Offset: 0

N. J. A. Sloane, Simon Plouffe

196 is conjectured to be the smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended the trajectory of 196 to millions of digits without finding a palindrome.
From A.H.M. Smeets, Jan 31 2019: (Start)
Palindromes for a(9)/2, a(14)/2 and a(20)/2.
Observed: It seems that most, but not all, Lychrel numbers (seeds given in A063048) have a trajectory term that, divided by 2, becomes palindromic. Note that 196 is the first Lychrel number (A063048(1)). (End)
Observed: On average, 0.414 digits are gained by each step of the reverse and add procedure; i.e., 2.416 steps are needed on average to gain a factor of 10. This holds for any trajectory of reverse and add for decimal number representation. - A.H.M. Smeets, Feb 03 2019

			From _M. F. Hasler_, Apr 13 2019: (Start)
Start with 196 = a(0), then:
A056964(196) = 196 + 691 = 887 = a(1); then:
A056964(887) = 887 + 788 = 1675 = a(2); then:
A056964(1675) = 1675 + 5761 = 7436 = a(3); then:
A056964(7436) = 7436 + 6347 = 13783 = a(4); then:
A056964(13783) = 13783 + 38731 = 52514 = a(5); etc. (End)

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 196, p. 58, Ellipses, Paris 2008.
D. H. Lehmer, "Sujets d'étude. No. 74," Sphinx (Bruxelles), 8 (1938), 12-13. (This is the currently earliest known reference to the 196 Problem). - James D. Klein, Apr 09 2012
Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 70.
Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975), pages PC30-6 to PC30-9.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Michael Lee, Table of n, a(n) for n = 0..2390 (T. D. Noe supplied terms 0 to 200)
Patrick De Geest, Some thematic websources
Jason Doucette, World Records
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012. - From _N. J. A. Sloane_, Nov 08 2012
Felix Fröhlich, C++ program for this sequence
Fred Gruenberger, How to handle numbers with thousands of digits, and why one might want to, Computer Recreations, Scientific American, 250 (No. 4, 1984), 19-26.
R. K. Guy, What's left?, Math Horizons, Vol. 5, No. 4 (April 1998), pp. 5-7.
Tim Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing
Madras Math's Amazing Number Facts, The Ultimate Palindrome
I. Peter, More trajectories
Wade VanLandingham, 196 and Other Lychrel Numbers
John Walker, Three Years Of Computing: Final Report On The Palindrome Quest
Eric Weisstein's World of Mathematics, 196-Algorithm.
Eric Weisstein's World of Mathematics, Palindromic Number Conjecture.
Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A033665, A016016, A056964, A004086.

a006960 n = a006960_list !! n
a006960_list = iterate a056964 196 -- Reinhard Zumkeller, Sep 22 2011

a:= proc(n) option remember; `if`(n=0, 196, (h-> h+ (s->
      parse(cat(s[-i]$i=1..length(s))))(""||h))(a(n-1)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 25 2014

a = {196}; For[i = 2, i < 26, i++, a = Append[a, a[[i - 1]] + ToExpression[ StringReverse[ToString[a[[i - 1]]]]]]]; a
NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&,196,25] (* Harvey P. Dale, Jun 05 2011 *)
NestList[#+IntegerReverse[#]&,196,25] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 04 2019 *)

A006960_vec(N=99)=vector(N,i,N=if(i>1,A056964(N),196)) \\ M. F. Hasler, Apr 13 2019

a(n+1) = A056964(a(n)). - A.H.M. Smeets, Jan 27 2019

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002

A033865 Start with n; if palindrome, stop; otherwise add to itself with digits reversed; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 121, 22, 33, 22, 55, 66, 77, 88, 99, 121, 121, 33, 44, 55, 33, 77, 88, 99, 121, 121, 363, 44, 55, 66, 77, 44, 99, 121, 121, 363, 484, 55, 66, 77, 88, 99, 55, 121, 363, 484, 1111, 66, 77, 88, 99, 121, 121, 66, 484, 1111, 4884, 77, 88, 99, 121, 121, 363, 484, 77, 4884, 44044, 88

Offset: 0

David W. Wilson

It is believed that a(196) = -1.

			19 -> 19 + 91 = 110 -> 110 + 011 = 121, so a(19) = 121.

M. Donner, I Love Me, Vol. I: S. Wordrow's palindromic encyclopedia (Algonquin Books, 1996) p. 268

T. D. Noe, Table of n, a(n) for n = 0..195
O. Forster, ARIBAS
Mathforum, Making Numbers into Palindromic Numbers
Eric Weisstein's World of Mathematics, 196-Algorithm.

Cf. A061563, A016016, A023109, A006960, A023108, A002113, A033665 (number of steps).

var st: stack; end; for k := 0 to 60 do n := k; while n <> int_reverse(n) do n := n + int_reverse(n); end; stack_push(st,n); end; stack2array(st);

Table[NestWhile[# + FromDigits[Reverse[IntegerDigits[#]]] &, n, IntegerDigits[#] != Reverse[IntegerDigits[#]] &], {n, 0, 90}] (* Harvey P. Dale, Dec 18 2011 *)

a(n)=my(k); while((k=fromdigits(Vecrev(digits(n)))) != n, n += k); n \\ infinite loop if a(n) = -1; Charles R Greathouse IV, Dec 13 2015

More terms from Jenise Smalley (neicey01(AT)hotmail.com), Oct 18 2001

A033665 Number of 'Reverse and Add' steps needed to reach a palindrome starting at n, or -1 if n never reaches a palindrome.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 1, 1, 1, 1, 2, 1, 0, 2, 3, 4, 1, 1, 1, 2, 1, 2, 2, 0, 4, 6, 1, 1, 2, 1, 2, 2, 3, 4, 0, 24, 1, 2, 1, 2, 2, 3, 4, 6, 24, 0, 1, 0, 1, 1

Offset: 0

N. J. A. Sloane

Palindromes themselves are not 'Reverse and Add!'ed, so they yield a zero!
Numbers n that may have a(n) = -1 (i.e., potential Lychrel numbers) appear in A023108. - Michael De Vlieger, Jan 11 2018
Record indices and values are given in A065198 and A065199. - M. F. Hasler, Feb 16 2020

			19 -> 19+91 = 110 -> 110+011 = 121 = palindrome, took 2 steps, so a(19)=2.
n = 89 needs 24 steps to end up with the palindrome 8813200023188. See A240510. - _Wolfdieter Lang_, Jan 12 2018

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers Penguin Books, 1987, pp. 142-143.

Kerry Mitchell, Table of n, a(n) for n = 0..195
P. De Geest, Some thematic websources
Jason Doucette, World Records
S. K. Eddins, The Palindromic Order Of A Number [archived page]
Kerry Mitchell, Table of n, a(n) for n = 0..10000 (The -1 entries are only conjectural)
Kerry Mitchell, Table of n, a(n) for n = 0..100000 (The -1 entries are only conjectural)
I. Peter, Search for the biggest numeric palindrome [archived page]
T. Trotter, Jr., Palindrome Power [archived page]
Eric Weisstein's World of Mathematics, 196-Algorithm.
Index entries for sequences related to Reverse and Add!

Cf. A002113, A023108, A023109, A033865, A006960, A016016, A063048, A240510.
Equals A030547(n) - 1.
Cf. A065198, A065199 (record indices & values).

rev[n_]:=FromDigits[Reverse[IntegerDigits[n]]];radd[n_]:=n+rev[n];
pal[n_]:=If[n==rev[n],True,False];
raddN[n_]:=Length[NestWhileList[radd[#]&,n,pal[#]==False&]]-1;
raddN/@Range[0,195] (* Ivan N. Ianakiev, Aug 31 2015 *)
With[{nn = 10^3}, Array[-1 + Length@ NestWhileList[# + IntegerReverse@ # &, #, !PalindromeQ@ # &, 1, nn] /. k_ /; k == nn -> -1 &, 200]] (* Michael De Vlieger, Jan 11 2018 *)

rev(n)={d=digits(n);p="";for(i=1,#d,p=concat(Str(d[i]),p));return(eval(p))}
a(n)=if(n==rev(n),return(0));for(k=1,10^3,i=n+rev(n);if(rev(i)==i,return(k));n=i)
n=0;while(n<100,print1(a(n),", ");n++) \\ Derek Orr, Jul 28 2014

A033665(n,LIM=333)={-!for(i=0,LIM,my(r=A004086(n)); n==r&&return(i); n+=r)} \\ with {A004086(n)=fromdigits(Vecrev(digits(n)))}. The second optional arg is a search limit that could be taken smaller up to very large n, e.g., 99 for n < 10^9, 200 for n < 10^14, 250 for n < 10^18: see A065199 for the records and A065198 for the n's. - M. F. Hasler, Apr 13 2019, edited Feb 16 2020

A033665 = lambda n, LIM=333: next((i for i in range(LIM) if is_A002113(n) or not(n := A004086(n)+n)), -1) # The second, optional argument is a search limit, see above. - M. F. Hasler, May 23 2024

More terms from Patrick De Geest, Jun 15 1998

I truncated the b-file at n=195, since the value of a(196) is not presently known (cf. A006960). The old b-files are now a-files. - N. J. A. Sloane, May 09 2015

A058042 Trajectory of binary number 10110 under the operation 'Reverse and Add!' carried out in base 2.

Original entry on oeis.org

10110, 100011, 1010100, 1101001, 10110100, 11100001, 101101000, 110010101, 1011101000, 1101000101, 10111010000, 11000101101, 101111010000, 110010001101, 1011110100000, 1100001011101, 10111110100000

Offset: 0

N. J. A. Sloane, May 18 2001

According to J. Walker, Ronald Sprague has proved that this trajectory does not contain a palindrome. [I would like a reference for this.] Another proof has been given by Klaus Brockhaus.
10110 is the smallest number with this property in base 2. The analogous number in base 10 is believed to be 196, but its trajectory (see A006960) has never been proved not to contain a palindrome.
The binary numbers have a regular pattern with cycle length 4:
  a(4k) = 101^(k+1)010^(k+1) for k >= 1,
  a(4k+1) = 1101^(k-1)0001^(k-1)01 for k >= 2,
  a(4k+2) = 101^(k+1)010^(k+2) for k >= 0,
  a(4k+3) = 110^(k+1)101^(k)01 for k >= 1, where ^ stands for repeated concatenation. - A.H.M. Smeets, Feb 03 2019
From A.H.M. Smeets, Feb 11 2019: (Start)
Pattern with cycle length 4 represented by contextfree grammars with production rules:
S_a -> 10 T_a 00, T_a -> 1 T_a 0 | 1101;
S_b -> 11 T_b 01, T_b -> 0 T_b 1 | 1000;
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | 1101;
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | 0010;
see also A058042 for similar grammars for the binary represented trajectory of 77. (End)

T. D. Noe, Table of n, a(n) for n = 0..500
T. Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing, Aug 22 1995.
Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
David J. Seal, Proofs similar to base 2 for base 4, 11, 17 and 26
J. Walker, Three Years Of Computing: Final Report On The Palindrome Quest
Index entries for sequences related to Reverse and Add!

See A061561 for the terms of A058042 written in base 10. Cf. A016016, A006960, A023108.

var m,c,rev: integer; end; m := 22; c := 1; bit_write(m); write(" "); rev := bit_reverse(m); while m <> rev and c < 25 do inc(c); m := m + rev; bit_write(m); write(" "); rev := bit_reverse(m); end;

a058042 = a007088 . a061561  -- Reinhard Zumkeller, Apr 21 2013

Clear[a]; a[0] = 10110; a[n_] := a[n] = (m = IntegerDigits[ a[n-1] ]; m2 = FromDigits[m, 2]; IntegerDigits[ FromDigits[m // Reverse, 2] + m2, 2] // FromDigits); Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 03 2013 *)

a(n) = A007088(A061561(n)). - Reinhard Zumkeller, Apr 21 2013

More terms from Klaus Brockhaus, May 27 2001

A065001 a(n) = (presumed) number of palindromes in the 'Reverse and Add!' trajectory of n, or -1 if this number is not finite.

Original entry on oeis.org

11, 10, 8, 9, 10, 7, 6, 8, 4, 9, 9, 6, 7, 5, 5, 7, 6, 3, 4, 8, 6, 8, 5, 5, 7, 6, 3, 4, 4, 6, 7, 5, 6, 7, 6, 3, 4, 4, 4, 7, 5, 5, 7, 7, 3, 4, 4, 4, 2, 5, 5, 7, 6, 3, 5, 4, 4, 2, 6, 5, 7, 6, 3, 4, 4, 5, 2, 6, 3, 7, 6, 3, 4, 4, 4, 2, 7, 3, 5, 6, 3, 4, 4, 4, 2, 6, 3, 6, 1, 3, 4, 4, 4, 2, 6, 3, 5, 1, 3, 8, 8, 6, 6

Offset: 1

Klaus Brockhaus, Nov 01 2001

Presumably a(196) = 0 (see A016016). Conjecture: There is no n > 0 such that the trajectory of n contains an infinite number of palindromes; the trajectory of n eventually leads to a term in the trajectory of some integer k which belongs to sequence A063048, i.e. whose trajectory (presumably) never leads to a palindrome.

			8, 77, 1111, 2222, 4444, 8888, 661166, 3654563 are the eight palindromes in the trajectory of 8 and 3654563 + 3654563 = 7309126 is the sixth term in the trajectory of 10577 (see A063433) which (presumably) never leads to a palindrome (see A063048), so a(8) = 8.

Index entries for sequences related to Reverse and Add!

A002113, A016016, A033865, A023108, A063048, A063433.

maxarg := 120; stop := 500; for k := 1 to maxarg do n := k; count := 0; c := 0; while c < stop do if n = int_reverse(n) then inc(count); c := 0; end; inc(c); n := n + int_reverse(n); end; write(count," " ); end;

A061563 Start with n; add to itself with digits reversed; if palindrome, stop; otherwise repeat; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 33, 55, 77, 99, 11, 22, 33, 44, 55, 66, 77, 88, 99, 121, 22, 33, 44, 55, 66, 77, 88, 99, 121, 121, 33, 44, 55, 66, 77, 88, 99, 121, 121, 363, 44, 55, 66, 77, 88, 99, 121, 121, 363, 484, 55, 66, 77, 88, 99, 121, 121, 363, 484, 1111, 66, 77, 88, 99, 121

Offset: 0

N. J. A. Sloane, May 18 2001

It is believed that n = 196 is the smallest integer which never reaches a palindrome.

			19 -> 19 + 91 = 110 -> 110 + 011 = 121, so a(19) = 121.

Cf. A033865. A016016 (number of steps), A023109, A006950, A023108.

var st: stack; test: boolean; end; for k := 0 to 60 do n := k; test := true; while test do n := n + int_reverse(n); test := n <> int_reverse(n); end; stack_push(st,n); end; stack2array(st);

tol = 1000; r[n_] := FromDigits[Reverse[IntegerDigits[n]]]; palQ[n_] := n == r[n]; ar[n_] := n + r[n]; Table[k = 0; If[palQ[n], n = ar[n]; k = 1]; While[! palQ[n] && k < tol, n = ar[n]; k++]; If[k == tol, n = -1]; n, {n, 0, 64}] (* Jayanta Basu, Jul 11 2013 *)
Table[Module[{k=n+IntegerReverse[n]},While[k!=IntegerReverse[k],k=k+IntegerReverse[k]];k],{n,0,70}] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Jul 19 2016 *)

Corrected and extended by Klaus Brockhaus, May 20 2001

More terms from Ray Chandler, Jul 25 2003

A345112 a(n) is the number of steps to reach a palindrome > n under repeated applications of the map x -> A345111(x) starting with n, or 0 if no palindrome is ever reached.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 19, 1, 1, 1, 1, 1, 1, 3, 1, 19

Offset: 1

Felix Fröhlich, Jun 09 2021

Is a(49) = 0? If a(49) != 0, it is > 100000 (10^5) (see A345115).

			For n = 39: The trajectory of 39 under the given map starts 39, 132, 453, 987, 1866, 10527, 15798, 73779, 111576, 227337, 500709, 507804, 585849, 1444344, 5887785, 14765640, 62422041, 86642457, 153067035, 683737386, reaching the palindrome 683737386 after 19 iterations, so a(39) = 19.

Cf. A016016, A345110, A345111, A345113, A345114, A345115.

eva(n) = subst(Pol(n), x, 10)
rot(vec) = if(#vec < 2, return(vec)); my(s=concat(Str(2), ".."), v=[]); s=concat(s, Str(#vec)); v=vecextract(vec, s); v=concat(v, vec[1]); v
a(n) = my(x=n, i=0); while(1, x=x+eva(rot(digits(x))); i++; if(digits(x)==Vecrev(digits(x)), break)); i

def pal(s): return s == s[::-1]
def rotl(s): return s[1:] + s[0]
def A345111(n): return n + int(rotl(str(n)))
def a(n):
    i, iter, seen = 0, n, set()
    while not (iter > n and pal(str(iter))) and iter not in seen:
        seen.add(iter)
        i, iter = i+1, A345111(iter)
    return i if iter > n and pal(str(iter)) else 0
print([a(n) for n in range(1, 49)]) # Michael S. Branicky, Jun 09 2021

A222533 Smallest palindrome which requires at least n iterations of Reverse and Add to reach a palindrome.

Original entry on oeis.org

1, 5, 77, 181, 747, 88, 494, 292, 4664, 9779, 464, 10901, 4884, 1991, 474, 999, 838, 64946, 989, 448844, 48084, 99199, 484, 73137, 48184, 94149, 11911, 390093, 46264, 567765, 26262, 28782, 3541453, 831138, 1447441, 493394, 1309031, 46664, 8817188, 17871

Offset: 1

Arkadiusz Wesolowski, Feb 24 2013

Palindromes themselves are also 'Reverse and Add!'ed!

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..110
Eric Weisstein's World of Mathematics, Reverse-Then-Add Sequence
Index entries for sequences related to palindromes
Index entries for sequences related to Reverse and Add!

Cf. A016016, A033665.

lst = {}; rev[n_] := FromDigits@Reverse@IntegerDigits[n]; testQ[n_] := ! rev[n] == n; Do[n = 1; While[True, If[! testQ[n], p = n + rev[n]; If[Length@TakeWhile[NestList[# + rev[#] &, p, s - 1], testQ] == s - 1, AppendTo[lst, n]; Break[]]]; n++], {s, 40}]; lst

A253241 The "Reverse and Add!" problem in base 12: sequence lists the final palindrome number for n, or -1 if no palindrome is ever reached. (Written in base 10.)

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 13, 39, 65, 91, 117, 143, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 169, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 169, 169, 39, 52, 65, 78, 91, 104, 117, 130, 143, 169, 169, 507, 52, 65, 78, 91, 104, 117, 130, 143, 169, 169, 507, 676, 65, 78, 91, 104, 117

Offset: 0

Eric Chen, Apr 07 2015

Is a(n) = -1 possible? All numbers below 100 (decimal 144) reach a palindrome.
a(237) is conjectured to be -1.
A060382 lists the smallest possible Lychrel number in base n.

			a(29) = 91 since (in duodecimal) 25 (decimal 29) + 52 = 77 (decimal 91) and 77 is a palindrome.
a(69) = 507 since (in duodecimal) 59 (decimal 69) + 95 = 132, 132 + 231 = 363 (decimal 507) and 363 is a palindrome.
a(105) = 1885 since (in duodecimal) 89 (decimal 105) + 98 = 165, 165 + 561 = 706, 706 + 607 = 1111 (decimal 1885) and 1111 is a palindrome.

Cf. A061563, A062129, A016016, A060382, A029957.

tol = 1728; r[n_] := FromDigits[Reverse[IntegerDigits[n, 12]], 12]; palQ[n_] := n == r[n]; ar[n_] := n + r[n]; Table[k = 0; If[palQ[n], n = ar[n]; k = 1]; While[! palQ[n] && k < tol, n = ar[n]; k++]; If[k == tol, n = -1]; n, {n, 0, 144}]

A333523 Number of iterations of Reverse And Add needed to reach a number divisible by n (or 0 if such a number is never reached).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 61, 1, 3, 8, 34, 22, 8, 17, 2, 8, 119, 14, 1, 17, 7, 110, 7, 12, 33, 34, 158, 28, 12, 1, 60, 11, 12, 50, 79, 7, 129, 64, 13, 42, 1, 4, 89, 131, 8, 14, 81, 30, 19, 125, 12, 1, 88, 13, 33, 67, 232, 26, 27, 24, 123, 59, 1, 24, 59, 36, 206, 148, 12, 217, 90, 97

Offset: 1

Daniel Starodubtsev, Mar 26 2020

If n is a palindrome > 0, a(n) = 1. See A002113.

a(n) > 0 for n < 10000.

			a(12) = 3, because 12 takes 3 iterations (12 -> 33 -> 66 -> 132) to become 132, which is divisible by 12.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A002113, A016016, A056964.

radd(n) = fromdigits(Vecrev(digits(n)))+n; \\ A056964
a(n) = {my(i=1, k=n, x); while((x=radd(n)) % k, i++; n=x); i;} \\ Michel Marcus, Apr 11 2020

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A006960 Reverse and Add! sequence starting with 196.

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A033865 Start with n; if palindrome, stop; otherwise add to itself with digits reversed; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

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A033665 Number of 'Reverse and Add' steps needed to reach a palindrome starting at n, or -1 if n never reaches a palindrome.

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A058042 Trajectory of binary number 10110 under the operation 'Reverse and Add!' carried out in base 2.

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A065001 a(n) = (presumed) number of palindromes in the 'Reverse and Add!' trajectory of n, or -1 if this number is not finite.

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A061563 Start with n; add to itself with digits reversed; if palindrome, stop; otherwise repeat; a(n) gives palindrome at which it stops, or -1 if no palindrome is ever reached.

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