A346175 a(0)=0. If a(n) is a novel term, a(n+1)=a(a(n)). If a(n) has appeared before, a(n+1)=number of prior terms equal to a(n).

0, 0, 1, 0, 2, 1, 1, 2, 1, 3, 0, 3, 1, 4, 2, 2, 3, 2, 4, 1, 5, 1, 6, 1, 7, 2, 5, 1, 8, 1, 9, 3, 3, 4, 2, 6, 1, 10, 0, 4, 3, 5, 2, 7, 1, 11, 3, 6, 2, 8, 1, 12, 1, 13, 4, 4, 5, 3, 7, 2, 9, 1, 14, 2, 10, 1, 15, 2, 11, 1, 16, 3, 8, 2, 12, 1, 17, 2, 13, 1, 18, 4, 6

Offset: 0

Author

David James Sycamore and Michael De Vlieger, Jul 08 2021

Keywords

• nonn
• look

Comments

This sequence is possible only with offset i=0 or 1, and a(i)=i. This is the i=0 version; variant i=1, beginning 1,1,1,2,1,3,1,4,2,1,... has similar properties.

Let conditions 0 and 1 respectively pertain to novel a(n) and extant a(n) respectively. When a(n) is novel, condition 0 gives a(n+1)=a(a(n)), a number which by definition is already extant. If a(n) has appeared k (>1) times already then by condition 1, a(n+1)=k-1. Following novel term N, a chain of repeat terms is generated by consecutive applications of condition 1, until reaching novel term N+1, whereupon a(N+1) initiates the next run of familiar terms. A novel term begets repeats of extant terms until these in turn produce another novel term, and so on; the sequence is infinite. No consecutive novel terms can be adjacent, there is at least one extant term between them, namely a(N)=1 gives N,1,N+1. The subsequence of records is the nonnegative integers appearing in order. Every number appears once, and thereafter arises infinitely many times.

a(m)=1 is the most frequently occurring term (arising both when a(m) is novel, with a(a(m))=1, and also as a(k+1) when a(k) is the second occurrence of m). The least occurring term is a(m)=0, which arises with increasing rarity as the sequence develops; namely at m= 0, 1, 2, 3, 10, 38, 225, 3635, 257942, ...

The subsequence {a(m+1): a(m) = record; 0,1,2,3,4,...} is identical to the original, so this sequence properly contains infinitely many copies of itself.

If condition 1 is replaced by the "count-back" rule of the Van Eck sequence (A181391), the result is A025480.

The plot of this sequence looks like sailboats on a lake; see links for details.

Examples

			a(0)=0 is a novel term so a(1)=a(a(0))=a(0)=0.
a(2)=1 because 0 has occurred twice.
Now a(2) is novel so a(3)=a(a(2))=a(1)=0, and so on.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Michael De Vlieger, Scatterplot of 2^18 terms showing "sailboat" shaped voids. The voids have 2 observed varieties, i.e., with 2 or 3 initial cusps.
• Michael De Vlieger, Scatterplot of 2^10 terms showing partition of the sequence into subsequences by records (gray gridlines) and "supersequences" by the duplication of a term (labeled).
• Michael De Vlieger, Enlarged scatterplot of a(1415..1651) between the records 123 and 138 which repeats the term 18 (i.e., supersequence 18 which includes subsequences 123..137.) This is a "sailboat" shape with 2 initial cusps.
• Michael De Vlieger, Enlarged scatterplot of a(1651..1898) between the records 138 and 151 which repeats the term 19 (i.e., supersequence 19 which includes subsequences 138..150.) This is a "sailboat" shape with 3 initial cusps.

Crossrefs

Cf. A025480, A181391.

Programs

Mathematica

Block[{a, c, j, k, r, nn}, nn = 120; c[_] := 0; a[0] = r = j = 0;
Do[If[a[n - 1] > r,
  r = a[n - 1]; k = a[a[n - 1]],
  k = c[a[n - 1]] ];
  c[a[n - 1]]++;
  Set[{a[n], j}, {k, k}], {n, nn}];
Array[a, nn, 0] ] (* Michael De Vlieger, May 25 2025 *)

Python

from itertools import islice
from collections import Counter
def agen(): # generator of terms
    an, c, r, alst = 0, Counter(), -1, []
    while True:
        yield an
        c[an] += 1
        alst.append(an)
        if an > r:
            an, r = alst[an], an
        else:
            an = c[an] - 1
print(list(islice(agen(), 83))) # Michael S. Branicky, May 25 2025