A364935 a(1) = 1 and thereafter a(n) = sum of digits of the number from concatenation of a(n-1) on the left with leftmost digit of a(n-1) + k and concatenation on the right with rightmost digit of a(n-1) + k, where k is the number of times a(n-1) has appeared in the sequence.

1, 5, 17, 18, 20, 6, 20, 8, 26, 18, 13, 10, 4, 14, 12, 8, 10, 6, 22, 10, 8, 12, 10, 10, 12, 12, 14, 14, 16, 16, 18, 15, 14, 18, 17, 20, 10, 14, 20, 12, 16, 20, 14, 13, 12, 18, 19, 13, 14, 15, 16, 13, 16, 15, 18, 21, 8, 14, 17, 13, 18, 23, 12, 20, 16, 17, 15, 20, 18, 25, 16, 19, 15, 13

Offset

1,2

Comments

Empirical observations: this sequence is extremely slow to grow, but I believe it diverges; the mode M(n) grows slowly and after some initial stabilization increases in intervals of 2; the distribution of even numbers in the limit follows a Gaussian distribution, and the distribution of odd numbers in the limits follows its *own* smaller Gaussian distribution.

The graph of the first 100000 terms looks somewhat like that 90's Jazz design that was on cups and other items, with a darker band concentrated in the middle of thick, diffuse swooshes.

Links

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Wikipedia, Jazz (design).

Example

For n=2, a(1) = 1 and "1" has appeared once (k=1) in the sequence. We add 1 to the leftmost digit, 1, and the rightmost digit, 1, and concatenate on each side so 212 which has sum of digits a(2) = 5.
For n=3, similarly a(2)=5 has appeared once (1) so we add 1 to the leftmost digit, 5, and the rightmost digit, 5, so 656 and its sum of digits is a(3) = 17.
For n=4, 17 -> 2178 -> 18, a(4) = 18.

Prog

(Python)
from collections import Counter
def A364935_list(max_n):
    A, C = [1], Counter()
    for n in range(2, max_n+1):
        x = str(A[n-2])
        C.update({x})
        z = str(int(x[0])+C[x]) + x + str(int(x[-1])+C[x])
        A.append(sum(int(z[i]) for i in range (0, len(z))))
    return(A) # John Tyler Rascoe, Oct 22 2023

Crossrefs

Sequence in context: A316548 A275629 A351974 * A249589 A040153 A038896

Adjacent sequences: A364932 A364933 A364934 * A364936 A364937 A364938

Keyword

base,easy,look,nonn

Author

Jarrod G. Sage, Sep 15 2023

Status

approved