A306806 An irregular fractal sequence: underline a(n) iff [a(n-1) + a(n)] is divisible by the rightmost digit of a(n); all underlined terms rebuild the starting sequence.

1, 2, 1, 3, 4, 2, 1, 5, 6, 3, 7, 8, 4, 2, 1, 9, 12, 13, 14, 15, 5, 16, 17, 18, 6, 3, 19, 22, 24, 23, 25, 26, 27, 28, 7, 32, 8, 4, 2, 1, 29, 33, 34, 35, 36, 9, 37, 38, 12, 39, 42, 43, 44, 13, 45, 46, 14, 47, 48, 49, 52, 54, 53, 55, 15, 5, 56, 16, 57

Offset

1,2

Comments

The sequence S starts with a(1) = 1 and a(2) = 2. No term is allowed to end with a 0 digit. S is extended by duplicating the first term A among the not yet duplicated terms, under the condition that the sum [a(n-1) + a(n)] is divisible by the rightmost digit of a(n). If this is not the case, we then extend S with the smallest integer X not yet present in S such that the sum [a(n-1) + a(n)] is not divisible by the rightmost digit of a(n). S is the lexicographically earliest sequence with this property.

Links

Jean-Marc Falcoz, Table of n, a(n) for n = 1..10002

Example

S starts with a(1) = 1 and a(2) = 2
Can we duplicate a(1) to form a(3)? Yes, as the sum [a(2) + a(3) = 3] is divisible by 1, the rightmost digit of a(3); thus a(3) = 1.
Can we duplicate a(2) to form a(4)? No, as the sum [a(3) + a(4) = 3] is not divisible by 2, the rightmost digit of a(4); we thus extend S with the smallest integer X not yet in S such that the sum [a(3) + X] is not divisible by the rightmost digit of a(4); thus a(4) = 3.
Can we duplicate a(2) to form a(5)? No, as the sum [a(4) + a(5) = 7] is not divisible by 2, the rightmost digit of a(5); we thus extend S with the smallest integer X not yet in S such that the sum [a(4) + X] is not divisible by the rightmost digit of a(5); thus a(5) = 4.
Can we duplicate a(2) to form a(6)? Yes, as the sum [a(5) + a(6) = 6 is divisible by 2, the rightmost digit of a(6); thus a(6) = 2.
Can we duplicate a(3) to form a(7)? Yes, as 1 can always be duplicated; thus a(7) = 1.
Can we duplicate a(4) to form a(8)? No, as the sum [a(7) + a(8) = 5] is not divisible by 2, the rightmost digit of a(8); we thus extend S with the smallest integer X not yet in S such that the sum [a(7) + X] is not divisible by the rightmost digit of a(8); thus a(8) = 5.
Can we duplicate a(4) to form a(9)? No, as the sum [a(8) + a(9) = 8] is not divisible by 3, the rightmost digit of a(9); we thus extend S with the smallest integer X not yet in S such that the sum [a(8) + X] is not divisible by the rightmost digit of a(9); thus a(9) = 6.
Can we duplicate a(4) to form a(10)? Yes, as the sum [a(9) + a(10) = 9] is divisible by 3, the rightmost digit of a(10); thus a(10) = 3.
Etc.

Crossrefs

Cf. A306805 [obtained by replacing the word _rightmost_ by _leftmost_ in the definition. This sequence diverges from A306805 with a(17) = 12, as opposite to a(17) = 20].

Sequence in context: A112384 A248514 A123390 * A306805 A162598 A088208

Adjacent sequences: A306803 A306804 A306805 * A306807 A306808 A306809

Keyword

base,nonn

Author

Eric Angelini and Jean-Marc Falcoz, Mar 11 2019

Status

approved