A243238 -id:A243238 - cp's OEIS frontend

A001127 Trajectory of 1 under map x->x + (x-with-digits-reversed).

1, 2, 4, 8, 16, 77, 154, 605, 1111, 2222, 4444, 8888, 17776, 85547, 160105, 661166, 1322332, 3654563, 7309126, 13528163, 49710694, 99312488, 187733887, 976071668, 1842242347, 9274664828, 17559329557, 93151725128, 175304440267, 937348843838, 1775697687577

Offset: 0

N. J. A. Sloane, Jun 05 2002

Normally one stops as soon as a palindrome is reached.
A Reverse and Add! sequence.
Trajectories of 25, 34, 43, 52, 59, 61, 68, 70, 86, 95, ..., merge into this sequence. - Robert G. Wilson v, Dec 16 2005

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 251 terms from Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

Cf. A006960, A023108, A033648, A033649, A033650, A033651.
Cf. A056964, A004086.
Row n=1 of A243238.

a001127 n = a001127_list !! n
a001127_list = iterate a056964 1 -- Reinhard Zumkeller, Sep 22 2011

a:= proc(n) option remember; `if`(n=0, 1, (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, Jun 18 2014

NestList[ # + FromDigits@Reverse@IntegerDigits@# &, 1, 30] (* Robert G. Wilson v, Dec 16 2005 *)
NestList[#+IntegerReverse[#]&,1,30] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 19 2019 *)

A033648 Trajectory of 3 under map x->x + (x-with-digits-reversed).

Original entry on oeis.org

3, 6, 12, 33, 66, 132, 363, 726, 1353, 4884, 9768, 18447, 92928, 175857, 934428, 1758867, 9447438, 17794887, 96644658, 182289327, 906271608, 1712444217, 8836886388, 17673772776, 85401510447, 159803020905, 668823329856, 1327746658722, 3606313135953, 7201626272016

Offset: 0

N. J. A. Sloane

Normally one stops as soon as a palindrome is reached.

A Reverse and Add! sequence.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 251 terms from Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

Cf. A006960, A001127, A033649, A033650, A033651.
Cf. A056964, A004086.
Row n=3 of A243238.

a033648 n = a033648_list !! n
a033648_list = iterate a056964 3 -- Reinhard Zumkeller, Sep 22 2011

a:= proc(n) option remember; `if`(n=0, 3, (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, Jun 18 2014

NestList[ # + FromDigits@Reverse@IntegerDigits@# &, 3, 29] (* Robert G. Wilson v, Dec 16 2005 *)

A281301 Trajectory of 1000004999700144385 under the "Reverse and Add!" operation.

Original entry on oeis.org

1000004999700144385, 6834415079694144386, 13668830049399288772, 41457129443403175403, 71914259877895350817, 143719619755790592734, 581014717313707510075, 1151030424627424920260, 1771324671891665221771, 3542550333873429453542, 5996099577656760005995

Offset: 0

Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 21 2017

1000004999700144385 is the largest of the first 225 numbers that require exactly 259 steps to turn into a palindrome (see A281390). The sequence reaches a 119-digit palindrome after 259 steps (see b-file). The number was obtained empirically using computer algorithms and was not reported before.

Row 1000004999700144385 of the array in A243238. - Felix Fröhlich, Jan 21 2017

			a(1) = 1000004999700144385 + 5834410079994000001 = 6834415079694144386.

Popular Computing (Calabasas, CA), The 196 Problem, Vol. 3 (No. 30, Sep 1975).

Sergei D. Shchebetov, Table of n, a(n) for n = 0..259
Jason Doucette, World Records
Yutaka Nishiyama, Numerical Palindromes and the 196 Problem, International Journal of Pure and Applied Mathematics, Volume 80, No. 3, 2012, 375-384.
R. Styer, The Palindromic Conjecture and the Fibonacci Sequence, Villanova University, 1986, 1-11.
C. W. Trigg, Palindromes by Addition, Mathematics Magazine, 40 (1967), 26-28.
C. W. Trigg, More on Palindromes by Reversal-Addition, Mathematics Magazine, 45 (1972), 184-186.
Wikipedia, Lychrel Number
196 and Other Lychrel Numbers, 196 and Lychrel Number
Index entries for sequences related to Reverse and Add!

Cf. A023109, A033672, A065198, A065199, A065320, A065321, A065322, A065323, A065324, A065325, A065326, A065327, A070743, A072216, A072217, A072218, A243238, A281390.

NestList[#+IntegerReverse[#]&,1000004999700144385,10] (* Harvey P. Dale, Dec 24 2021 *)

terms(n) = my(x=1000004999700144385, i=0); while(1, print1(x, ", "); x=x+eval(concat(Vecrev(Str(x)))); i++; if(i==n, break))
/* Print initial 9 terms as follows: */
terms(9) \\ Felix Fröhlich, Jan 21 2017

a(n+1) = a(n) + rev(a(n)).

a(9)-a(10) from Felix Fröhlich, Jan 21 2017

A033650 Trajectory of 7 under map x --> x + (x-with-digits-reversed).

Original entry on oeis.org

7, 14, 55, 110, 121, 242, 484, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688, 17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933, 47267087164, 93445163438, 176881317877, 955594506548, 1801200002107

Offset: 0

N. J. A. Sloane

A Reverse and Add! sequence.
Trajectories of 19, 23, 28, 29, 32, 37, 38, 41, 46, 47, 49, 50, ..., merge into this sequence. - Robert G. Wilson v, Dec 16 2005
A Reverse and Add! sequence.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 251 terms from Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

Cf. A006960, A001127, A033648, A033649, A033651.
Cf. A056964, A004086.
Row n=7 of A243238.

a033650 n = a033650_list !! n
a033650_list = iterate a056964 7 -- Reinhard Zumkeller, Sep 22 2011

a:= proc(n) option remember; `if`(n=0, 7, (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, Jun 18 2014

NestList[ # + FromDigits@Reverse@IntegerDigits@# &, 7, 26] (* Robert G. Wilson v *)

A033651 Trajectory of 9 under map x->x + (x-with-digits-reversed).

Original entry on oeis.org

9, 18, 99, 198, 1089, 10890, 20691, 40293, 79497, 158994, 658845, 1207701, 2284722, 4559544, 9019098, 17928207, 88211178, 175322466, 839546037, 1570191975, 7362102726, 13634115363, 49985258994, 99970517988, 188942025987, 978462275868, 1847034540747, 9317488848228

Offset: 0

N. J. A. Sloane

Trajectories of 27, 36, 45, 54, 63, 72, 81, 90, ..., merge into this sequence. - Robert G. Wilson v, Dec 16 2005.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 251 terms from Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

Cf. A006960, A001127, A033648, A033649, A033650.
Cf. A056964, A004086.
Row n=9 of A243238.
Partial sums of A112296.

a063051 n = a063051_list !! n
a063051_list = iterate a056964 879 -- Reinhard Zumkeller, Sep 22 2011

a:= proc(n) option remember; `if`(n=0, 9, (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, Jun 18 2014

NestList[ # + FromDigits@Reverse@IntegerDigits@# &, 9, 26] (* Robert G. Wilson v *)

A033649 Trajectory of 5 under map x->x + (x-with-digits-reversed).

Original entry on oeis.org

5, 10, 11, 22, 44, 88, 176, 847, 1595, 7546, 14003, 44044, 88088, 176176, 847847, 1596595, 7553546, 14007103, 44177144, 88354288, 176599676, 853595347, 1597190705, 6668108656, 13236127322, 35608290553

Offset: 0

N. J. A. Sloane

Trajectories of 15, 21, 24, 30, 39, 42, 48, 51, 57, 60, 69, 75, 78, 84, 87, 93, 96, ..., merge into this sequence. - Robert G. Wilson v, Dec 16 2005
A Reverse and Add! sequence.
Trajectories of 13, 17, 20, 26, 31, 35, 40, 53, 62, 71, 79, 80, 97, ..., merge into this sequence. - Robert G. Wilson v, Dec 16 2005

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 251 terms from Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

Cf. A006960, A001127, A033648, A033650, A033651.
Cf. A056964, A004086.
Row n=5 of A243238.

a033649 n = a033649_list !! n
a033649_list = iterate a056964 5 -- Reinhard Zumkeller, Sep 22 2011

a:= proc(n) option remember; `if`(n=0, 5, (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, Jun 18 2014

NestList[ # + FromDigits@Reverse@IntegerDigits@# &, 5, 29] (* Robert G. Wilson v, Dec 16 2005 *)

A033652 Trajectory of 13 under map x->x + (x-with-digits-reversed).

Original entry on oeis.org

13, 44, 88, 176, 847, 1595, 7546, 14003, 44044, 88088, 176176, 847847, 1596595, 7553546, 14007103, 44177144, 88354288, 176599676, 853595347, 1597190705, 6668108656, 13236127322, 35608290553, 71117571206

Offset: 0

N. J. A. Sloane

To get the next term, reverse the present term and add it to itself.

			44 = 13 + 31.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Reverse and Add!

Row n=13 of A243238.

NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&,13,30] (* Harvey P. Dale, Mar 04 2015 *)

a(n+1) = a(n) + {a(n) reversed}.

A244058 n-th term of the 'Reverse and Add!' sequence starting with n.

Original entry on oeis.org

1, 4, 12, 77, 44, 363, 484, 4444, 79497, 14003, 88088, 175857, 1596595, 1716517, 17794887, 13528163, 176599676, 839546037, 1317544822, 853595347, 8836886388, 13236127322, 13297007933, 668823329856, 175304440267, 909153350908, 9317488848228, 8813200023188

Offset: 1

Alois P. Heinz, Jun 18 2014

a(n) is a palindrome for n in {1, 2, 4, 5, 6, 7, 8, 9, 11, 21, 28, 30}.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to Reverse and Add!

Main diagonal of A243238.

Cf. A001127, A033648, A033670, A006960, A004086.

b:= proc(n, j) option remember; `if`(j=1, n, (h-> h+ (s->
      parse(cat(s[-i]$i=1..length(s))))(""||h))(b(n, j-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..40);

Table[Nest[#+IntegerReverse[#]&,n,n-1],{n,30}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 07 2021 *)

A298972 Number of positive integers k < n such that n occurs in the Reverse-and-Add trajectory of k.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 2, 2, 0, 1, 0, 4, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Felix Fröhlich, Jan 30 2018

Number of integers k < n such that n occurs in row k of A243238.
For n > 0, a(n) = 0 iff n is a term of A067031.
For n > 0, a(n) > 0 iff n is a term of A067030.

			For n = 22: There exist 4 positive integers k < 22 such that 22 occurs in the Reverse-and-Add trajectory of k, namely 5, 10, 11 and 20, so a(22) = 4.

Index entries for sequences related to Reverse and Add!

Cf. A056964, A067030, A067031, A243238.

Block[{nn = 85, s}, s = Array[Union@ NestWhileList[# + IntegerReverse@ # &, #, # < nn &, 1, nn] &, nn]; Array[Count[Take[s, # - 1], #, 2] &, nn + 1, 0]] (* Michael De Vlieger, Feb 01 2018 *)

a(n) = my(i=0); for(k=1, n-1, my(x=k); while(x < n, x=x+eval(concat(Vecrev(Str(x))))); if(x==n, i++)); i

A277338 Reverse and Add! sequence starting with 295.

Original entry on oeis.org

295, 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, 84724043932847, 159547977975595

Offset: 0

Matt C. Anderson, Oct 09 2016

Apart from the initial term in both sequences, the same as A006960.
a(0) = 295; a(n+1) = a(n) + A004086(a(n)).
295 is conjectured to be the second smallest initial term which does not lead to a palindrome. Also, 196 is possibly the smallest initial term which does not lead to a palindrome. a(0) = 196 is described in A006960.

			a(0) = 295
a(1) = 295 + 592 = 887
a(2) = 887 + 788 = 1675
...

Wikipedia, Lychrel number

Cf. A004086.
Almost the same as A006960.
See index entries at A023108.

with(StringTools):
revnum := proc (n)
local a, b, c;
description "to REVerse the digits of a NUMber";
a := convert(n, string);
b := Reverse(a);
c := convert(b, decimal, 10)
end proc;
f := 0;
e := 295;
count := 0;
while f <> e do
e := e+f;
f := revnum(e);
count := count+1
end do;

a[1] = 295; a[n_] := a[n] = FromDigits@ Reverse@ IntegerDigits@ # + # &@ a[n - 1]; Array[a, 29] (* Michael De Vlieger, Oct 14 2016 *)

terms(n) = my(x=295, i=0); while(1, print1(x, ", "); x=x+eval(concat(Vecrev(Str(x)))); i++; if(i==n, break))
/* Print initial 30 terms as follows: */
terms(30) \\ Felix Fröhlich, Nov 15 2016

a(n) = A006960(n) for n >= 1.

a(n) = A243238(295, n+1). - Felix Fröhlich, Nov 20 2016

cp's OEIS Frontend

A001127 Trajectory of 1 under map x->x + (x-with-digits-reversed).

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A033648 Trajectory of 3 under map x->x + (x-with-digits-reversed).

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A281301 Trajectory of 1000004999700144385 under the "Reverse and Add!" operation.

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A033650 Trajectory of 7 under map x --> x + (x-with-digits-reversed).

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A033651 Trajectory of 9 under map x->x + (x-with-digits-reversed).

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A033649 Trajectory of 5 under map x->x + (x-with-digits-reversed).

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A033652 Trajectory of 13 under map x->x + (x-with-digits-reversed).

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