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A166512 2-comma numbers: n occurs in the sequence S[k+1] = S[k] + 10*last_digit(S[k-1]) + first_digit(S[k]) for two different splittings n=concat(S[0],S[1]).

%I A166512 #21 Dec 23 2024 14:53:42
%S A166512 1023,1044,1521,1657,1789,1984,2191,2263,2451,2466,2523,2676,2783,
%T A166512 2824,3066,3268,3589,3602,3631,4051,4113,4149,4159,4213,4315,4611,
%U A166512 4685,4781,4969,5133,5526,6053,6165,6246,6445,6650,6712,6893,7350,7668,8011,8144
%N A166512 2-comma numbers: n occurs in the sequence S[k+1] = S[k] + 10*last_digit(S[k-1]) + first_digit(S[k]) for two different splittings n=concat(S[0],S[1]).
%C A166512 A comma number n as defined in A166511 is the concatenation of numbers a,b (no leading zeros allowed) which occurs ("again") in the sequence S = S(a,b) defined by S[0]=a, S[1]=b, S[k+1] = S[k] + 10*last_digit(S[k-1]) + first_digit(S[n]).
%C A166512 Here we list the subsequence of numbers that can be split up in 2 different ways, n=concat(a,b)=concat(c,d), such that S(a,b) and S(c,d) both contain n.
%C A166512 Since the 4-digit terms remind of year numbers, the terminology of bicommatile (in analogy with bissextile) years has been introduced (as a joke).
%H A166512 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 A166512 E. Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/Commatile.htm">k-comma numbers</a>, Oct. 2009.
%H A166512 E. Angelini, <a href="/A166507/a166507.pdf">k-comma numbers</a> [Cached copy, with permission]
%e A166512 None of the 3-digit terms in A166511 can be split up in 2 ways such that S(a,bc) and S(ab,c) both contain n=abc (concatenation, not product).
%e A166512 Therefore the smallest term in this sequence is a(1)=1023, which occurs in the sequences S(102,3) and S(10,23).
%o A166512 (PARI) {for(n=1,1e4,/*is_A166512(n)=*/ my(c=2); 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 & c--==0 & /*return(1)*/ !print1(n", ") & break))}
%Y A166512 Cf. A166507, A166508, A166511, A166513.
%K A166512 base,easy,nonn
%O A166512 1,1
%A A166512 _Eric Angelini_ and _M. F. Hasler_, Oct 28 2009