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%I #135 Feb 07 2024 13:19:52
%S 1,12,35,94,135,186,248,331,344,387,461,475,530,535,590,595,651,667,
%T 744,791,809,908,997,1068,1149,1240,1241,1252,1273,1304,1345,1396,
%U 1457,1528,1609,1700,1701,1712,1733,1764,1805,1856,1917,1988,2070
%N 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.
%C 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.
%C The sequence contains exactly 2137453 terms, with a(2137453)=99999945. The next term does not exist. - _W. Edwin Clark_, Dec 11 2006
%C It is remarkable that the sequence persists for so long. - _N. J. A. Sloane_, Dec 15 2006
%C The similar sequence A139284, which starts at a(1)=2, persists even longer, ending at a(194697747222394) = 9999999999999918. - _Giovanni Resta_, Nov 30 2019
%C Conjecture: This sequence is finite, for any initial term. - _N. J. A. Sloane_, Nov 14 2023
%C 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
%C Does not satisfy Benford's Law. - _Michael S. Branicky_, Nov 16 2023
%C 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
%D Eric Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
%H Michael S. Branicky, <a href="/A121805/b121805.txt">Table of n, a(n) for n = 1..20000</a> (terms 1..1001 from Zak Seidov)
%H Eric Angelini, <a href="/A121805/a121805.pdf">The Commas Sequence</a>, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]
%H Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, <a href="http://arxiv.org/abs/2401.14346">arXiv:2401.14346</a>, <a href="https://www.youtube.com/watch?v=_EHAdf6izPI">Youtube</a>
%H Lorenzo Angelini, <a href="https://www.youtube.com/watch?v=_Se0yJSbD48">Happy birthday Éric!!</a>, Youtube video.
%H Michael S. Branicky, <a href="/A121805/a121805.png">Graph of a(n)/n over entire sequence</a>
%H Simon Demers, <a href="https://drive.google.com/uc?export=download&id=16cV-LPDX_pBels0Doj69J14IXBRRNq2I">Table of n, a(n) for n = 1..2137453</a> (full sequence)
%H Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_980.htm">Puzzle 980. The "Commas" sequence</a>, The Prime Puzzles and Problems Connection.
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%e 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:
%e Sequence: 1, 12, 35, 94, 135, 186, 248, 331, 344, 387, 461, 475, ...
%e Differences: 11, 23, 59, 41 , 51 , 62 , 83 , 13 , 43 , 74 , 14 , ...
%e To illustrate the formula in the comment: a(6) = 186 and a(7) = 248 = 186 + 62.
%p digits:=n->ListTools:-Reverse(convert(n,base,10)):
%p 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:
%p A121805:=proc(n) option remember: if n = 1 then return 1; fi; return nextK(A121805(n-1)); end: # _W. Edwin Clark_
%t 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 *)
%o (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
%o (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
%o (Python)
%o from itertools import islice
%o def agen(): # generator of terms
%o an, y = 1, 1
%o while y < 10:
%o yield an
%o an, y = an + 10*(an%10), 1
%o while y < 10:
%o if str(an+y)[0] == str(y):
%o an += y
%o break
%o y += 1
%o print(list(islice(agen(), 45))) # _Michael S. Branicky_, Apr 08 2022
%Y See A366487 and A367349 for first differences.
%Y 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.
%Y Cf. A330128, A330129, A367338 (comma-successor), A367360.
%Y See also A260261, A042948.
%K nonn,base,fini,nice
%O 1,2
%A _Eric Angelini_, Dec 11 2006
%E More terms from _Zak Seidov_, Dec 11 2006
%E Edited by _N. J. A. Sloane_, Sep 17 2023
%E Changed name from "commas sequence" to "comma sequence". - _N. J. A. Sloane_, Dec 20 2023
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