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A358921 a(1) = 1; a(n) is the smallest positive number not among the terms a(n-c .. n-1) where c = the number of times a(n-1) has occurred.

%I A358921 #25 Jan 13 2023 09:19:35
%S A358921 1,2,1,3,1,2,3,1,4,1,5,1,2,3,4,1,6,1,7,1,5,2,3,4,1,8,1,9,1,6,2,3,4,1,
%T A358921 5,2,6,1,7,2,3,4,5,1,8,2,6,3,7,1,9,2,4,5,3,6,1,10,1,11,1,12,1,13,1,14,
%U A358921 1,8,2,3,4,5,6,1,7,2,9,1,15,1,16,1,17,1
%N A358921 a(1) = 1; a(n) is the smallest positive number not among the terms a(n-c .. n-1) where c = the number of times a(n-1) has occurred.
%C A358921 A new number other than 1 is always followed by a 1, so a(n) < n/2 for n > 4.
%H A358921 Samuel Harkness, <a href="/A358921/b358921.txt">Table of n, a(n) for n = 1..10000</a>
%H A358921 Samuel Harkness, <a href="/A358921/a358921.jpg">Scatterplot of the first 2000000 terms</a>
%e A358921 For a(6), a(5) = 1 has occurred 3 times, so the smallest positive integer not in {a(5), a(4), a(3)} = {1, 3, 1} is 2, thus a(6) = 2.
%e A358921 Next, for a(7), a(6) = 2 has occurred 2 times, so the smallest positive integer not in {a(6), a(5)} = {2, 1} is 3, thus a(7) = 3.
%e A358921 Then, for a(8), a(7) = 3 has occurred 2 times, so the smallest positive integer not in {a(7), a(6)} = {3, 2} is 1, thus a(8) = 1.
%e A358921 Now, for a(9), a(8) = 1 has occurred 4 times, so the smallest positive integer not in {a(8), a(7), a(6), a(5)} = {1, 3, 2, 1} is 4, thus a(9) = 4.
%e A358921 The first terms, alongside the number of times they have occurred o(n), are:
%e A358921   n  a(n)  o(n)
%e A358921   -  ----  ----
%e A358921   1     1     1
%e A358921   2     2     1
%e A358921   3     1     2
%e A358921   4     3     1
%e A358921   5     1     3
%e A358921   6     2     2
%e A358921   7     3     2
%e A358921   8     1     4
%e A358921   9     4     1
%e A358921   10    1     5
%t A358921 V = {1} While[Length[V] < 84, b = 1; While[MemberQ[Take[V, -Count[V, Last[V]]], b], b++ ]; AppendTo[V, b]]; Print[V]
%o A358921 (PARI) { a = o = vector(84); v = 1; for (n=1, #a, print1 (a[n]=v", "); v=setminus([1..n+1], Set(a[n-o[a[n]]+++1..n]))[1]) } \\ _Rémy Sigrist_, Jan 09 2023
%Y A358921 Cf. A133622, A268696, A329985.
%K A358921 nonn,hear,look
%O A358921 1,2
%A A358921 _Samuel Harkness_, Dec 06 2022