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A325775
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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.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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Cf. A308306 (the "boomerang numbers") and A325776 where duplicate terms are admitted.
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KEYWORD
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sign,base
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AUTHOR
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STATUS
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approved
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