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Sequences of RADD type (for: "reverse digits, then add") are defined by a recurrence of the form ${\displaystyle a_{n+1}=R(a_{n})+Q}$ where R = A004086 is digit reversal and Q is a fixed constant.

## Introduction

Let T(S,Q) be the "RADD" sequence formed by starting with S, and then extending by the rule: reverse digits (i.e., apply function A004086) and add Q.

Leading zeros are omitted, of course.

This family was proposed by Luc Stevens (lms022(AT)yahoo.com) in an email to N. J. A. Sloane, Apr 05 2006.

David Applegate, Klaus Brockhaus and several other people have also contributed to this work.

The general behavior is that for given values of S and Q, either:

1. T(S,Q) will after a finite number (i, say) of steps enter a cycle of finite length (c, say), or else
2. T(S,Q) will never reach a cycle (in which case we set i = c = -1).

Example: S = Q = 1. This reaches a cycle of length 9 in one step:

```1, 2,3,4,5,6,7,8,9,10, 2,3,4,5,6,7,8,9,10, 2,3,4,5,6,7,8,9,10, 2,3,4,5,6,7,... (see A117230).
```

Example: S = 1, Q = 10. This never reaches a cycle:

```1,11,21,22,32,33,43,44,54,55,65,66,76,77,87,88,98,99,109,911,129,931,149,... (see A117841)
```

## Summary

S=1, Q=n: steps to reach cycle = A117816, length of cycle = A117817

S=n, Q=1: steps to reach cycle = A118511, records A118510 (cycle length c is always 9)

S=n, Q=2: steps to reach cycle = A118514, records A118515, A118516 (cycle length c = 81 for many small values of S)

S=n, Q=3: steps to reach cycle = A118522, records A118523, A118524 (cycle length c = 3 or 6 for many small values of S)

S=n, Q=4: steps to reach cycle = A117831, records A118473, A118474 (cycle length c = 54 for S=1..1014, see A117830, A117827, A117807)

## Entries for sequences T(S,Q)

```  S   Q   i      c   A-number
-----------------------------
A117816 A117817
1   1   1      9   A117230
1   2   1     81   A117521
1   3   1      3   A118517
1   4   1     54   A117828 See A117830, A117827 for cycle
1   5   1    207   A117800
1   6   1     30   A118525
1   7   1     63   A118526
1   8   1     27   A118527
1   9   1      1   (boring)
1  10  -1     -1   A117841
1  11   1      9   A118528
1  12   2     15   A118529
1  13  31     18   A118530
1  14  15     72   A118531
1  15  -1     -1   A118532
1  16 721     90   A118533
1  17   9     54   A118606
1  18   1     13   A118607
1  19   6     18   A118608
1  20  -1     -1   A118535
1  21   3     15   A118602
1  22   5      9   A118603
1  23  28      9   A118609
1  24  29     36   A118610
1  25 131     45   A118543
1  26  23     18   A118615
1  27   1      9   A118613
1  28  31     36   A118614
1  29   6     18   A118616
1  30  -1     -1   A118637
1  31   1      9   A118617
1  32  19      9   A118618
1  33   1      3   A118619
1  34  53     36   A118631
1  35   4     72   A118632
1  36 406      2   A118536
1  37  34     27   A118633
1  38 122     18   A118634
1  39   8      3   A118635
1  40  -1     -1   A118636
1  43   1     63   A118087
1  45   2     22   A118620
1  50  -1     -1   A118147
1  60  -1     -1   A118162
1  70  -1     -1   A118217
```
```  S   Q   i      c   A-number
-----------------------------
A118511
(records: A118510)
1   1   1      9   A117230
2   1   0      9   A117230
3   1   0      9   A117230
4   1   0      9   A117230
5   1   0      9   A117230
6   1   0      9   A117230
7   1   0      9   A117230
8   1   0      9   A117230
9   1   0      9   A117230
10   1   0      9   A117230
11   1  18      9   A118512
12   1  17      9   A118512
13   1  15      9   A118513
```
```  S   Q   i      c   A-number
-----------------------------
A118522
(records: A118523, A118524)
1   3   1      3   A118517
2   3   3      6   A118518
3   3   3      6   A118519
4   3   0      3   A118517
5   3   2      6   A118520
6   3   2      6   A118521
7   3   0      3   A118517
```
```  S   Q   i      c   A-number
-----------------------------
A117831
(records: A118473, A118474)
1   4   1     54   A117828 See A117830, A117827 for cycle
2   4   1     54   see A117828
3   4  40     54   A117829
4   4   7     54
5   4   0     54   A117828
6   4   0     54   A117828
7   4  39     54   A117829
8   4   6     54   A117829
1015  4   0     90   A117807
```

A binary version from Jan Nordbotten: A118360