Sequences of the RADD type -------------------------- N. J. A. Sloane, May 07 2006 Let T(S,Q) be the "RADD" sequence formed by starting with S, and then extending by the rule: reverse digits and add Q. Leading zeros are omitted, of course. This family was proposed by Luc Stevens (lms022(AT)yahoo.com) in an email to njas, 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: (i) T(S,Q) will after a finite number (i, say) of steps enter a cycle of finite length (c, say), or else (ii) 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 93 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 (end)