A033670 -id:A033670 - cp's OEIS frontend

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

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 *)

A240510 Numbers whose "reverse and add" process becomes palindromic at 8813200023188.

Original entry on oeis.org

89, 98, 187, 286, 385, 583, 682, 781, 869, 880, 968, 1297, 1387, 1477, 1567, 1657, 1747, 1837, 1927, 2296, 2386, 2476, 2566, 2656, 2746, 2836, 2926, 3295, 3385, 3475, 3565, 3655, 3745, 3835, 3925, 4294, 4384, 4474, 4564, 4654, 4744, 4834, 4924, 5293, 5383

Offset: 1

J. Lowell, Apr 06 2014

The "reverse and add" sequence for 178 includes 8813200023188, but it becomes palindromic at 15851, so 178 is not in this sequence.

See interesting patterns of the first differences in A328492. - Robert Price, Oct 16 2019

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 143.

Daniel Starodubtsev, Table of n, a(n) for n = 1..100000 (terms 1..177 from Robert Price)
Felix Fröhlich, C++ program for this sequence.

Cf. A033670, A328492.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
Select[Range[0, 50000], (np = #; i = 0;
   While[ ! PalindromeQ[np] && i < limit,
    np = np + IntegerReverse[np]; i++];
np == 8813200023188) &] (* Robert Price, Oct 16 2019 *)

is(n)=my(k=8813200023188); while(n<=k && (d=digits(n))!=(r=Vecrev(d)), n+=fromdigits(r)); n==k; \\ Charles R Greathouse IV, Apr 09 2020

More terms from Jon E. Schoenfield, Apr 12 2014

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 *)

A328492 First differences of A240510 (Numbers whose "reverse and add" process becomes palindromic at 8813200023188).

Original entry on oeis.org

9, 89, 99, 99, 198, 99, 99, 88, 11, 88, 329, 90, 90, 90, 90, 90, 90, 90, 369, 90, 90, 90, 90, 90, 90, 90, 369, 90, 90, 90, 90, 90, 90, 90, 369, 90, 90, 90, 90, 90, 90, 90, 369, 90, 90, 90, 90, 90, 90, 90, 369, 90, 90, 90, 90, 90, 90, 90, 369, 90, 90, 90, 90

Offset: 1

Robert Price, Oct 16 2019

Additional patterns can be seen in the bfile.

Robert Price, Table of n, a(n) for n = 1..176

Cf. A033670, A240510.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
Differences@Select[Range[0, 50000], (np = #; i = 0;
    While[ ! PalindromeQ[np] && i < limit,
     np = np + IntegerReverse[np]; i++];
np == 8813200023188) &] (* Robert Price, Oct 16 2019 *)

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

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A240510 Numbers whose "reverse and add" process becomes palindromic at 8813200023188.

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

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A328492 First differences of A240510 (Numbers whose "reverse and add" process becomes palindromic at 8813200023188).

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