cp's OEIS Frontend

A166511 Comma numbers: n=concat(a,b) occurs in the sequence S[0]=a, S[1]=b, S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]).

%I A166511 #15 Dec 23 2024 14:53:42
%S A166511 10,20,30,40,50,60,70,80,90,109,123,148,160,163,185,198,199,241,340,
%T A166511 362,398,422,423,444,522,540,541,550,564,597,621,622,667,683,693,724,
%U A166511 769,770,780,806,811,835,842,888,972,1023,1044,1053,1054,1116,1146,1177
%N A166511 Comma numbers: n=concat(a,b) occurs in the sequence S[0]=a, S[1]=b, S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]).
%C A166511 A comma number n is the concatenation of numbers a,b (no leading zeros allowed) which occurs ("again") in the comma sequence S = S(a,b) defined by S[0]=a, S[1]=b, S[n+1] = S[n] + 10*last_digit(S[n-1]) + first_digit(S[n]), i.e., add to a given term the number formed by the two digits surrounding the preceding comma.
%C A166511 The sequence S is infinite and straightforward to compute, in contrast to the implicitly defined terms of A121805.
%C A166511 The sequence S(a,b) is strictly increasing, unless a=0 (mod 10) and b=0 (which implies n=0 (mod 100)), in which case all following terms are zero.
%H A166511 E. Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2009-October/002530.html">Comma numbers</a>, SeqFan mailing list, Oct 15 2009
%H A166511 E. Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/Commatile.htm">k-comma numbers</a>, Oct. 2009.
%H A166511 E. Angelini, <a href="/A166507/a166507.pdf">k-comma numbers</a> [Cached copy, with permission]
%e A166511 a(1)=10 is in the sequence, because splitting 10 into (1,0) gives the sequence S(1,0) = 1, 0, 0+10=10, 10+01=11, 11+01=12, 12+11=23,... which contains 10.
%e A166511 a(10)=109 is in the sequence, because splitting 109 into (10,9) gives the sequence S(10,9) = 10, 9, 9+09=18, 18+91=109, 109+81=190,... which contains 109.
%e A166511 108 is not in the sequence since S(10,8) does not contain 108 and it is not allowed to split 108 -> 1,08 -> S(1,08).
%o A166511 (PARI) {for(n=1,1e4,/*iscomma(n)=*/for(d=1,#Str(n)-1, my( a=n\10^d, b=n%10^d ); b<10^(d-1) & (d>1 || a%10==0) & next; while(n > b=10*(a%10)+b\10^(#Str(b)-1)+a=b,); b>n & next; /*return(1)*/ print1(n", "); break))}
%Y A166511 See also subsequences A166508, containing A166509, and A166512, containing A166513.
%K A166511 base,nonn
%O A166511 1,1
%A A166511 _Eric Angelini_ and _M. F. Hasler_, Oct 28 2009