A357256 "Forest Fire" sequence with the additional condition that no progression of the form ABA is allowed for any terms A and B.

1, 1, 2, 2, 4, 4, 5, 3, 3, 5, 6, 6, 7, 10, 10, 7, 9, 12, 11, 9, 12, 8, 8, 14, 14, 11, 15, 13, 13, 17, 23, 20, 16, 15, 17, 23, 24, 16, 18, 18, 19, 26, 21, 28, 25, 19, 22, 22, 29, 24, 20, 30, 27, 21, 32, 29, 30, 35, 26, 34, 36, 25, 31, 32, 34, 37, 39, 36, 28, 27

Offset: 1

Author

Neal Gersh Tolunsky, Dec 11 2022

Keywords

• nonn

Comments

It is easy to see that a number can occur no more than twice: 1) If a number occurs twice, one term with that value must be at an odd n and the other at an even n. This is because otherwise you could always find a progression of the form ABA. 2) Once two terms of the same value are in the sequence on an even and odd n, no third term with that value can be added without creating a progression of form ABA.

Examples

			a(4)=2 because if a(4) were 1 the 2-4th terms would be the ABA-form progression 1,2,1. 2 here is the smallest number which forms neither an arithmetic nor ABA progression.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A229037.

Programs

Python

from itertools import count, islice
def agen(): # generator of terms
    alst, mink, aba = [0], [1, 1], [set(), set()] # even, odd appearances
    for n in count(1):
        k = mink[n&1]
        ff = set(2*alst[n-i] - alst[n-2*i] for i in range(1, (n+1)//2))
        while k in ff or k in aba[n&1]: k += 1
        alst.append(k); aba[n&1].add(k); yield k
        while mink[n&1] in aba[n&1]: mink[n&1] += 1
print(list(islice(agen(), 70))) # Michael S. Branicky, Dec 12 2022

Extensions

More terms from Michael S. Branicky, Dec 12 2022