A378381 - OEIS

%I #23 Nov 28 2024 11:26:22

%S 1,1,2,3,2,4,3,5,6,4,7,3,8,9,5,10,6,11,12,7,13,3,14,15,16,8,17,9,18,5,

%T 19,20,10,21,6,22,23,11,24,12,25,26,13,27,3,28,29,30,31,14,32,15,33,

%U 16,34,8,35,36,17,37,9,38,39,18,40,5,41,42,43,19,44,20,45

%N Lexicographically earliest sequence such that each set of terms enclosed by two equal values, including the endpoints, contains a distinct number of elements.

%C The word 'set' means that every element is unique. For example, the set {1,1,2} contains 2 elements (not 3).

%C Note that we are considering sets between every pair of equal values, not just those that appear consecutively.

%C Two consecutive values enclose 1 term, and thus after [a(1), a(2)] = [1, 1], no consecutive equal values occur again.

%H Neal Gersh Tolunsky, <a href="/A378381/b378381.txt">Table of n, a(n) for n = 1..10000</a>

%e a(4) cannot be 1 since this would create a second pair enclosing two values, [1,2,1] being an equivalent set to [1,2,1,1]. We cannot have a(4)=2 because [1,2,1] would enclose the same number of elements as [2,1,2]. So a(4)=3, which has not occurred before.

%o (Python)

%o from itertools import islice

%o def agen(): # generator of terms

%o     e, a = set(), []

%o     while True:

%o         an, allnew = 0, False

%o         while not allnew:

%o             allnew, an, ndset = True, an+1, set()

%o             for i in range(len(a)):

%o                 if an == a[i]:

%o                     nd = len(set(a[i:]))

%o                     if nd in e or nd in ndset: allnew = False; break

%o                     ndset.add(nd)

%o         yield an; a.append(an); e |= ndset

%o print(list(islice(agen(), 73))) # _Michael S. Branicky_, Nov 26 2024

%Y Cf. A366691.

%K nonn

%O 1,3

%A _Neal Gersh Tolunsky_, Nov 26 2024