A033648 -id:A033648 - cp's OEIS frontend

A145734 Indices of palindromes in A033648.

1, 2, 4, 5, 7, 10, 23, 32

Offset: 1

Dmitry Kamenetsky, Oct 17 2008

The next term (if it exists) is greater than 10000.

These are actually indices into A033648 right-shifted; A033648 has offset 0, but this sequence treats it as having offset 1.

A033648

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

Original entry on oeis.org

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 *)

A243238 Table T(n,r) of terms in the reverse and add sequences of positive integers n read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 4, 4, 3, 8, 8, 6, 4, 16, 16, 12, 8, 5, 77, 77, 33, 16, 10, 6, 154, 154, 66, 77, 11, 12, 7, 605, 605, 132, 154, 22, 33, 14, 8, 1111, 1111, 363, 605, 44, 66, 55, 16, 9, 2222, 2222, 726, 1111, 88, 132, 110, 77, 18, 10, 4444, 4444, 1353, 2222, 176, 363, 121, 154, 99, 11, 11

Offset: 1

Felix Fröhlich, Jun 12 2014

			T(5,6) = 88, since 88 is the 6th term in the reverse and add sequence of 5.
Table starts with:
  1 2 4 8 16 77 154 605 1111 2222
  2 4 8 16 77 154 605 1111 2222 4444
  3 6 12 33 66 132 363 726 1353 4884
  4 8 16 77 154 605 1111 2222 4444 8888
  5 10 11 22 44 88 176 847 1595 7546
  6 12 33 66 132 363 726 1353 4884 9768
  7 14 55 110 121 242 484 968 1837 9218
  8 16 77 154 605 1111 2222 4444 8888 17776
  9 18 99 198 1089 10890 20691 40293 79497 158994
  10 11 22 44 88 176 847 1595 7546 14003

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index entries for sequences related to Reverse and Add!

Rows n=1, 3, 5, 7, 9 give: A001127, A033648, A033649, A033650, A033651.
Cf. A006960, A023108, A033670, A088753, A089694, A240510.
Main diagonal gives A244058.

T:= proc(n, r) option remember; `if`(r=1, n, (h-> h +(s->
      parse(cat(s[-i]$i=1..length(s))))(""||h))(T(n, r-1)))
    end:
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Jun 18 2014

rad[n_] := n + FromDigits[Reverse[IntegerDigits[n]]];
T[n_, 1] := n; T[n_, k_] := T[n, k] = rad[T[n, k-1]];
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Apr 08 2016 *)

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 *)

A244567 Triangle T(n,k) in which the n-th row lists in increasing order all values s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s; triangle T(n,k), n>=0, 1<=k<=A244569(n), read by rows.

Original entry on oeis.org

0, 1, 1, 2, 3, 1, 2, 4, 5, 3, 6, 7, 1, 2, 4, 8, 9, 5, 10, 5, 10, 11, 3, 6, 12, 13, 7, 14, 15, 1, 2, 4, 8, 16, 17, 9, 18, 19, 20, 21, 5, 10, 11, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 3, 6, 12, 21, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43

Offset: 0

Alois P. Heinz, Jun 30 2014

			Triangle T(n,k) begins:
   0;
   1;
   1,  2;
   3;
   1,  2,  4;
   5;
   3,  6;
   7;
   1,  2,  4,  8;
   9;
   5, 10;
   5, 10, 11;
   3,  6, 12;
  13;
   7, 14;
  15;
   1,  2,  4,  8,  16;
  17;

Alois P. Heinz, Rows n = 0..6000, flattened

Column k=0 gives A244568.
Last elements of rows give A001477.
Cf. A001127, A033648, A033670, A006960, A004086, A244569.

A244568 Smallest value s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 5, 3, 13, 7, 15, 1, 17, 9, 19, 20, 21, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 3, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 5, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 7, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 3, 67, 68, 69

Offset: 0

Alois P. Heinz, Jun 30 2014

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Column k=0 of A244567.

Cf. A001127, A033648, A033670, A006960, A004086, A244569.

A244569 Number of values s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 3, 3, 1, 2, 1, 5, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17

Offset: 0

Alois P. Heinz, Jun 30 2014

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Row lengths of A244567.

Cf. A001127, A033648, A033670, A006960, A004086, A244568.

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 *)

cp's OEIS Frontend

A145734 Indices of palindromes in A033648.

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

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Maple

Mathematica

A243238 Table T(n,r) of terms in the reverse and add sequences of positive integers n read by antidiagonals.

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Maple

Mathematica

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

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Haskell

Maple

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

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Haskell

Maple

Mathematica

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

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Haskell

Maple

Mathematica

A244567 Triangle T(n,k) in which the n-th row lists in increasing order all values s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s; triangle T(n,k), n>=0, 1<=k<=A244569(n), read by rows.

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A244568 Smallest value s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s.

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A244569 Number of values s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s.

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