A325775 Numbers that bring back the last digit of n on its first digit, using the "boomerang protocol" explained in A308306, with no duplicate term (except the -1 terms). See the Comments and Example sections for details.

102, 25, 20, 27, 22, 29, 24, 90, 26, 0, 104, 10, 40, 30, 42, 49, 44, 70, 28, 3, 23, 5, 12, 7, 2, 9, 4, 19, 6, 14, 106, 21, 60, 32, 62, 47, 46, 50, 48, 13, 52, 15, 45, 17, 34, 37, 36, 39, 16, 38, 108, 41, 80, 43, 64, 54, 66, 69, 68, 33, 72, 35, 74, 55, 67, 57, 56, 59, 58, 18, 128, 61, 82, 63, 84, 65, 86, 76, 88, 53, 92, 73, 94, 75, 96

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, May 20 2019

Keywords

• sign
• base

Comments

All the nonnegative integers will appear in this sequence, except the terms of A308306.

The "boomerang protocol" sends 1 to the left (as 1 is odd - the even digits move to the right), jumping over exactly 1 cell. To "bring back" 1 to its initial cell, the smallest integer is 102. Let's see how:

Our initial 1 starts for instance here (dots are cells):

....1....

and ends there (S is the starting cell):

..1.S....

We have this pattern now for the "bring back" integer (S is the new start, A is the Arrival cell we must reach - which was the starting cell of 1):

..S.A....

The smallest integer starting on S and ending on A is 102:

..1.A....

0...A....

.2..A....

We see that 1 jumps to the left over 1 cell, 0 to the right over 0 cell (thus moving to this cell), 2 jumps over 2 cells and lands precisely on the Arrival cell.

Note that many integers can "bring back" 1 in its initial cell, 120 is one of them, for instance, or 1410.

Examples

			The sequence starts with 102,25,20,27,22,29,24,90,... We see that:
a(1) = 102 means that 102 will bring 1 back in its initial cell;
a(2) = 25 means that 25 will bring 2 back in its initial cell;
a(3) = 20 means that 20 will bring 3 back in its initial cell;
a(4) = 27 means that 27 will bring 4 back in its initial cell;
a(5) = 22 means that 22 will bring 5 back in its initial cell;
The general formula being that a(n) brings back (n) in its initial cell.
a(100) = -1 means that 100 is a "boomerang number": it "comes back" by itself without any external help. Those numbers are listed in A308306.
		

Links

• Jean-Marc Falcoz, Table of n, a(n) for n = 1..12000

Crossrefs

Cf. A308306 (the "boomerang numbers") and A325776 where duplicate terms are admitted.