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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 (list; graph; refs; listen; history; text; internal format)
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(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
REFERENCES
Eric Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
LINKS
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, Youtube
Lorenzo Angelini, Happy birthday Éric!!, Youtube video.
Simon Demers, Table of n, a(n) for n = 1..2137453 (full sequence)
Carlos Rivera, Puzzle 980. The "Commas" sequence, The Prime Puzzles and Problems Connection.
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:
A121805:=proc(n) option remember: if n = 1 then return 1; fi; return nextK(A121805(n-1)); end: # W. Edwin Clark
MATHEMATICA
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 *)
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")=", a-a%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[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
(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
print(list(islice(agen(), 45))) # Michael S. Branicky, Apr 08 2022
CROSSREFS
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.
Sequence in context: A368782 A367635 A367645 * A367359 A367344 A195542
KEYWORD
nonn,base,fini,nice
AUTHOR
Eric Angelini, Dec 11 2006
EXTENSIONS
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
STATUS
approved

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Last modified April 22 08:12 EDT 2024. Contains 371893 sequences. (Running on oeis4.)