

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.


71



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
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OFFSET

1,2


COMMENTS

An equivalent, but more formal definition, is: a(1) = 1; for n > 1, let x be the least significant digit of a(n1); then a(n) = a(n1) + 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
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


REFERENCES

Eric Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 3235, Volume 59 (Jeux math'), April/June 2008, Paris.


LINKS

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, Youtube


EXAMPLE

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.


MAPLE

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:


MATHEMATICA

a[1] = 1; a[n_] := a[n] = For[x=Mod[a[n1], 10]; y=0, y <= 9, y++, an = a[n1] + 10*x + y; If[y == IntegerDigits[an][[1]], Return[an]]]; Array[a, 45] (* JeanFrançois Alcover, Nov 25 2014 *)


PROG

(PARI) 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")=", aa%10*10)) \\ M. F. Hasler, Jul 21 2015
(R) 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[i1] %% 10; k = 1; while (k != as.integer(substring(A121805$a[i1]+LSD*10+k, 1, 1))){k = k+1; if(k>9) break} A121805$a[i]=A121805$a[i1]+LSD*10+k} # Simon Demers, Oct 19 2017
(Python)
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


CROSSREFS

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.


KEYWORD

nonn,base,fini,nice


AUTHOR



EXTENSIONS

Changed name from "commas sequence" to "comma sequence".  N. J. A. Sloane, Dec 20 2023


STATUS

approved



