A366487 First differences of "commas" sequence A121805.

11, 23, 59, 41, 51, 62, 83, 13, 43, 74, 14, 55, 5, 55, 5, 56, 16, 77, 47, 18, 99, 89, 71, 81, 91, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 1, 11, 21, 31, 41, 51, 61, 71, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 2, 22, 42, 62, 82, 2, 22, 42, 63, 93, 23, 53, 83, 13, 43, 73, 3, 33, 63, 93, 23, 53, 83, 13, 43, 73, 3, 33, 63, 94, 34, 74, 14, 54, 94, 34, 74, 14, 54, 94

Offset: 1

Author

N. J. A. Sloane, Nov 12 2023

Keywords

• nonn
• fini
• base

Comments

The record high points in this sequence are 11, 23, 59, 62, 83, 99, and they occur at terms 1, 2, 3, 6, 7, 21. Since 99 is the largest possible term, this is the full list of record high points.

The first differences of this sequence (i.e., the second differences of A121805) fall into the range [-90,90] for the first 99999 terms.

More generally, for a commas sequence in base b, the first differences are <= b^2 - 1. - Michael S. Branicky, Nov 16 2023.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..20000 (terms 1..1000 from N. J. A. Sloane)
• Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Fibonacci Quarterly 62:3 (2024), 215-232.
• Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
• Michael S. Branicky, Table of n, a(n) for n = 1..100000
• Michael S. Branicky, Graph of entire sequence
• N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides

Crossrefs

Cf. A121805.

For the numbers missing from this sequence see A367349.

Programs

Python

from itertools import islice
def agen(): # generator of terms
    an, y = 1, 1
    while y < 10:
        prevan = an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
        yield an - prevan
print(list(islice(agen(), 99))) # Michael S. Branicky, Nov 12 2023