A368782 Comma transform of A366487.

12, 35, 94, 15, 16, 28, 31, 34, 37, 41, 45, 55, 55, 55, 55, 61, 67, 74, 71, 89, 98, 97, 18, 19, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 11, 11, 12, 13, 14, 15, 16, 17, 18, 22, 22, 24, 26, 28, 22, 22, 24, 26, 28, 22, 22, 24, 26, 28, 22, 22, 24, 26, 28, 22, 22

Offset: 1

Author

Michael S. Branicky, Jan 05 2024

Keywords

• nonn
• base

Comments

See A367360 for further information.

Let the comma sequence A121805 be known as S or C0.

A366487, the first differences of A121805, is the same as the comma transform of A121805; call it C1.

This sequence is C2 = C(C(S)), the comma transform C iterated twice.

C4 = C2, C5 = C2, ... once the first term (and the last term if the sequence is finite) are removed from the lower iterates of C.

Theorem: C^{i+2}(S) = C^i(S) for i>=2 in general and for i>=0 when all terms of S have two digits and no least significant digit is zero. See link for proof.

Remark. The lexicographically earliest sequence S with C(S) = S is A010850, all 11's.

The sequence contains 2137451 terms, with a(2137451) = 96. The next term does not exist.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..20000
• Michael S. Branicky, Comma Comma Proof

Crossrefs

Cf. A010850, A121805, A367360, A366487.

Programs

Python

from itertools import islice, pairwise
def S(): # generator of comma sequence
    an = 1
    while True:
        yield an
        an += 10*(an%10)
        children = [an+y for y in range(1, 10) if str(an+y)[0] == str(y)]
        if not children: break
        an = children[0]
def C(g): # generator of comma transform of sequence passed as a generator
    yield from (10*(t%10) + int(str(u)[0]) for t, u in pairwise(g))
def agen(): return C(C(S()))
print(list(islice(agen(), 70))) # Michael S. Branicky, Jan 05 2024