A354971 a(1)=1, a(2)=0; for n > 2, a(n) is the number of times a(n - 1 - a(n-1)) has appeared in the sequence.

1, 0, 1, 1, 3, 1, 1, 5, 5, 5, 1, 3, 3, 3, 6, 3, 5, 5, 5, 5, 1, 7, 1, 1, 9, 5, 9, 8, 8, 9, 9, 1, 4, 2, 10, 4, 10, 4, 1, 3, 2, 11, 4, 11, 4, 2, 2, 5, 5, 2, 10, 5, 5, 12, 2, 12, 2, 7, 3, 2, 2, 7, 9, 2, 3, 3, 5, 3, 10, 10, 10, 10, 3, 7, 13, 4, 7, 7, 7, 7, 11

Offset

1,5

Comments

Conjecture: every positive integer eventually appears in the sequence.

Conjecture: The longest run of same, consecutive a(n) is 4 terms.

A003056 can be generated using the same rules but starting with 0.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16, showing records in red, first entries of other m in green, and the latest entries of m in blue.

Neal Gersh Tolunsky, Illustration of first 100 terms

Example

For n=9, a(9) is the number of times a(9 - 1 - a(9-1)) = a(8 - a(8)) = a(3) = 1 has appeared in the sequence, so a(9)=5.

Mathematica

nn = 2^20; c[_] = 0; a[1] = c[0] = c[1] = 1; a[2] = j = 0; Do[k = c[a[n - j - 1]]; a[n] = j = k; c[k]++, {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Jun 25 2022 *)

Prog

(MATLAB) function a = A354971( max_n )
    a = [1 0]; c = zeros(1, max_n); c(1:2) = [1 1];
    for n = 3:max_n
        m = a(n-1-a(n-1))+1;
        a = [a c(m)];
        c(c(m)+1) = c(c(m)+1)+1;
    end
end % Thomas Scheuerle, Jun 15 2022
(PARI) { nb = [0]; for (n=1, #a=vector(81), print1 (a[n] = if (n==1, 1, n==2, 0, nb[1+a[n-1-a[n-1]]])", "); if (#nb < 1+a[n], nb = concat(nb, vector(#nb))); nb[1+a[n]]++) } \\ Rémy Sigrist, Jun 18 2022
(Python)
from collections import Counter
from itertools import count, islice
def agen():
    a = [None, 1, 0]; inventory = Counter(a); yield from a[1:]
    for n in count(3):
        c = inventory[a[n-1-a[n-1]]]
        a.append(c); inventory.update([c]); yield c
print(list(islice(agen(), 81))) # Michael S. Branicky, Jun 18 2022

Crossrefs

Cf. A003056, A329985

Sequence in context: A124925 A073145 A245368 * A239331 A145033 A202672

Adjacent sequences: A354968 A354969 A354970 * A354972 A354973 A354974

Keyword

nonn,easy

Author

Neal Gersh Tolunsky, Jun 14 2022

Status

approved