A361429 a(n) is the smallest positive number not among the terms between a(n-1) and the most recent previous term whose value appears with the same frequency (inclusive); if no such term exists, set a(n)=1; a(1)=1.

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

Offset

1,3

Comments

From Samuel Harkness, Mar 11 2023: (Start)

Observations:

Record values k > 2 seem to occur at the following places:

First k for k == 0 (mod 3) occurs at n = 2^(k/3+2) + k/3 - 4;

First k for k == 1 (mod 3) occurs at n = 2^((k-1)/3+2) + (k-1)/3 - 3;

First k for k == 2 (mod 3) occurs at n = 3*(2^((k+1)/3)) + (k-14)/3.

For any value k, frequency(k) ~= 2*frequency(3+k). For any value j >= 0, frequency(2+j) ~= frequency(3+j) ~= frequency(4+j).

This sequence contains many recurring strings. For example, {1, 3, 4, 1, 2} occurs 12499 times in the first 100000 terms. From its 5th occurrence at a(40) through its 64th occurrence at a(517), the number of terms between each {1, 3, 4, 1, 2} minus one gives

{1 3 1 4 ...}

First 1 term of A001511, 3, first 1 term of A001511, 4.

{... 1 2 1 4 1 2 1 5 ...}

First 3 terms of A001511, 4, first 3 terms of A001511, 5.

{... 1 2 1 3 1 2 1 5 1 2 1 3 1 2 1 6 ...

First 7 terms of A001511, 5, first 7 terms of A001511, 6.

{... 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 6 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 7 ...}

First 15 terms of A001511, 6, first 15 terms of A001511, 7.

(End)

Links

Samuel Harkness, Table of n, a(n) for n = 1..10000

Samuel Harkness, MATLAB program

Example

a(6)=4 because in the sequence so far (1, 1, 2, 1, 3), the most recent term with the same number of occurrences as a(5)=3 is a(3)=2. Between a(3) and a(5), (2, 1, 3), the smallest missing number is 4, so a(6)=4.
a(8)=2 because between a(7)=1 and the previous value with the same frequency count a(4)=1 (1, 3, 4, 1), the smallest missing number is 2, so a(8)=2.

Prog

(MATLAB) See Links section.

Crossrefs

Cf. A361172, A358921, A001511.

Sequence in context: A147567 A247045 A347354 * A345290 A372619 A330669

Adjacent sequences: A361426 A361427 A361428 * A361430 A361431 A361432

Keyword

nonn

Author

Neal Gersh Tolunsky, Mar 11 2023

Status

approved