A367349 -id:A367349 - cp's OEIS frontend

A121805 The "comma sequence": the lexicographically earliest sequence of positive numbers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms is the same as the sequence of successive differences between the terms.

1, 12, 35, 94, 135, 186, 248, 331, 344, 387, 461, 475, 530, 535, 590, 595, 651, 667, 744, 791, 809, 908, 997, 1068, 1149, 1240, 1241, 1252, 1273, 1304, 1345, 1396, 1457, 1528, 1609, 1700, 1701, 1712, 1733, 1764, 1805, 1856, 1917, 1988, 2070

Offset: 1

Eric Angelini, Dec 11 2006

An equivalent, but more formal definition, is: a(1) = 1; for n > 1, let x be the least significant digit of a(n-1); then a(n) = a(n-1) + x*10 + y where y is the most significant digit of a(n) and is the smallest such y, if such a y exists. If no such y exists, stop.
The sequence contains exactly 2137453 terms, with a(2137453)=99999945. The next term does not exist. - W. Edwin Clark, Dec 11 2006
It is remarkable that the sequence persists for so long. - N. J. A. Sloane, Dec 15 2006
The similar sequence A139284, which starts at a(1)=2, persists even longer, ending at a(194697747222394) = 9999999999999918. - Giovanni Resta, Nov 30 2019
Conjecture: This sequence is finite, for any initial term. - N. J. A. Sloane, Nov 14 2023
The base 2 analog (suggested by William Cheswick) is 1, 4, 5, 8, 9, 12, 13, ..., (see A042948) with successive differences 3, 1, 3, 1, ... (repeat). - N. J. A. Sloane, Nov 15 2023
Does not satisfy Benford's Law. - Michael S. Branicky, Nov 16 2023
Using the notion of "comma transform" of a sequence, as defined in A367360, this is the lexicographically earliest sequence of positive integers with the property that its first differences and comma transform coincide. - N. J. A. Sloane, Nov 23 2023

			Replace each comma in the original sequence by the pair of digits adjacent to the comma; the result is the sequence of first differences between the terms of the sequence:
Sequence:   1, 12, 35, 94, 135, 186, 248, 331, 344, 387, 461, 475, ...
Differences: 11, 23, 59, 41 , 51 , 62 , 83 , 13 , 43 , 74 , 14 , ...
To illustrate the formula in the comment: a(6) = 186 and a(7) = 248 = 186 + 62.

Eric Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

Michael S. Branicky, Table of n, a(n) for n = 1..20000 (terms 1..1001 from Zak Seidov)
Eric Angelini, The Commas Sequence, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]
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.
Lorenzo Angelini, Happy birthday Éric!!, Youtube video.
Michael S. Branicky, Graph of a(n)/n over entire sequence
Simon Demers, Table of n, a(n) for n = 1..2137453 (full sequence)
Robert Dougherty-Bliss, The Comma Sequence is WILD, 2024 video.
Robert Dougherty-Bliss and Natalya Ter-Saakov, The Comma Sequence is Finite in Other Bases, arXiv:2408.03434 [math.NT], 2024.
Carlos Rivera, Puzzle 980. The "Commas" sequence, The Prime Puzzles and Problems Connection.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides
Index entries for sequences related to Benford's law

See A366487 and A367349 for first differences.
Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.
Cf. A330128, A330129, A367338 (comma-successor), A367360.
See also A260261, A042948.

digits:=n->ListTools:-Reverse(convert(n,base,10)):
nextK:=proc(K) local i,L; for i from 0 to 9 do L:=K+digits(K)[ -1]*10+i; if i = digits(L)[1] then return L; fi; od; FAIL; end:
A121805:=proc(n) option remember: if n = 1 then return 1; fi; return nextK(A121805(n-1)); end: # W. Edwin Clark

a[1] = 1; a[n_] := a[n] = For[x=Mod[a[n-1], 10]; y=0, y <= 9, y++, an = a[n-1] + 10*x + y; If[y == IntegerDigits[an][[1]], Return[an]]]; Array[a, 45] (* Jean-François Alcover, Nov 25 2014 *)

a=1; for(n=1,1000, print1(a", "); a+=a%10*10; for(k=1, 9, digits(a+k)[1]==k&&(a+=k)&&next(2)); error("blocked at a("n")=",a-a%10*10)) \\ M. F. Hasler, Jul 21 2015

from itertools import islice
def agen(): # generator of terms
    an, y = 1, 1
    while y < 10:
        yield an
        an, y = an + 10*(an%10), 1
        while y < 10:
            if str(an+y)[0] == str(y):
                an += y
                break
            y += 1
print(list(islice(agen(), 45))) # Michael S. Branicky, Apr 08 2022

A121805 <- data.frame(n=seq(from=1,to=2137453),a=integer(2137453)); A121805$a[1]=1; for (i in seq(from=2,to=2137453)){LSD=A121805$a[i-1] %% 10; k = 1; while (k != as.integer(substring(A121805$a[i-1]+LSD*10+k,1,1))){k = k+1; if(k>9) break} A121805$a[i]=A121805$a[i-1]+LSD*10+k} # Simon Demers, Oct 19 2017

More terms from Zak Seidov, Dec 11 2006
Edited by N. J. A. Sloane, Sep 17 2023
Changed name from "commas sequence" to "comma sequence". - N. J. A. Sloane, Dec 20 2023

A366487 First differences of "commas" sequence A121805.

Original entry on oeis.org

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

N. J. A. Sloane, Nov 12 2023

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.

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

Cf. A121805.

For the numbers missing from this sequence see A367349.

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

cp's OEIS Frontend

A121805 The "comma sequence": the lexicographically earliest sequence of positive numbers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms is the same as the sequence of successive differences between the terms.

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