A309681 Lexically least sequence such that a(n) next appears at index n+a(n+1) and such that a(1)=2.

2, 4, 5, 6, 2, 6, 4, 13, 5, 6, 2, 6, 13, 9, 5, 18, 2, 18, 9, 4, 19, 13, 9, 17, 6, 20, 18, 26, 30, 17, 13, 35, 5, 19, 2, 19, 35, 23, 4, 9, 47, 23, 17, 20, 6, 30, 50, 4, 60, 20, 50, 5, 18, 2, 18, 50, 41, 26, 23, 35, 7, 42, 17, 47, 21, 13, 35, 37, 60, 5, 19, 2, 19, 60, 6

Offset: 1

Author

Samuel B. Reid, Aug 12 2019

Keywords

• nonn

Comments

This sequence is aperiodic. Proof: If any number of initial terms are removed, said initial terms can be recreated by extrapolating backwards from later terms. In other words, the later terms include a perfect description of the earlier terms. This implies that, if the sequence is periodic, the repeating section must include all of the initial terms of the sequence in the order in which they originally appear. This is impossible because we cannot extrapolate backwards infinitely from the initial terms. If we attempt to do so, we get the pattern 5, 5, 4, 2, 4, 5 (the last three values being the first three terms of the sequence). This pattern is clearly invalid, so the sequence must be aperiodic. - Samuel B. Reid, Jun 07 2020

Examples

			If a(2) were 1, a(2) would have to be 2. a(2) cannot be both 1 and 2, so a(2) cannot be 1.
If a(2) were 2, the next 2 after a(1) would appear at a(3). The next 2 actually appears at a(2), so a(2) cannot be 2.
If a(2) were 3, a(4) would have to be 2. If a(4) were 2, a(3) and a(5) would have to have the same value. If a(3) and a(5) were equal, the pattern X, 3, Y, 2 would appear later in the sequence. The pattern X, 3, Y, 2 is invalid because X would have to be 2. X cannot be 2 because it comes after the 2 at a(1) and before the 2 that succeeds Y. This behavior is inconsistent with the definition of the sequence.
The next possible value for a(2) is 4, which causes no trivial inconsistencies (as with a(2)=1 or a(2)=2) or logical inconsistencies (as with a(2)=3).
The n-th term of the sequence is the lowest value that does not cause inconsistencies of either sort.
		

Links

• Samuel B. Reid, Table of n, a(n) for n = 1..10000
• Samuel B. Reid, Python program for A309681
• Samuel B. Reid, C program for A309681

Crossrefs

Cf. A171922.