cp's OEIS Frontend

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

%I A309681 #55 Jul 01 2020 23:44:08
%S A309681 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,
%T A309681 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,
%U A309681 18,50,41,26,23,35,7,42,17,47,21,13,35,37,60,5,19,2,19,60,6
%N A309681 Lexically least sequence such that a(n) next appears at index n+a(n+1) and such that a(1)=2.
%C A309681 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
%H A309681 Samuel B. Reid, <a href="/A309681/b309681.txt">Table of n, a(n) for n = 1..10000</a>
%H A309681 Samuel B. Reid, <a href="/A309681/a309681.txt">Python program for A309681</a>
%H A309681 Samuel B. Reid, <a href="/A309681/a309681.c.txt">C program for A309681</a>
%e A309681 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.
%e A309681 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.
%e A309681 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.
%e A309681 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).
%e A309681 The n-th term of the sequence is the lowest value that does not cause inconsistencies of either sort.
%o A309681 (Python and C) See Links section.
%Y A309681 Cf. A171922.
%K A309681 nonn
%O A309681 1,1
%A A309681 _Samuel B. Reid_, Aug 12 2019