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A166513 3-comma numbers: n occurs in the sequence S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]) for three different splittings n=concat(S[0],S[1]).

%I A166513 #12 Nov 11 2019 00:37:37
%S A166513 2676,6246,8349,9427,10587,11558,11756,11811,12427,12788,13090,13110,
%T A166513 14328,15031,15187,15493,15637,16867,18322,18768,19918,20699,21138,
%U A166513 21422,21698,22824,23108,23242,23868,24456,24854,25342,25478,26583
%N A166513 3-comma numbers: n occurs in the sequence S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]) for three different splittings n=concat(S[0],S[1]).
%C A166513 This subsequence of A166512 consists of the numbers that can be split up in (at least) three different ways, n=concat(a,b)=concat(c,d)=concat(e,f), such that the sequences S(a,b), S(c,d) and S(e,f) all contain n.
%C A166513 (Here S(a,b) is the sequence defined by S[0]=a, S[1]=b, S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]).) See A166511 and A166512 for more information.
%H A166513 E. Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/Commatile.htm">k-comma numbers</a>, Oct. 2009.
%H A166513 E. Angelini, <a href="/A166507/a166507.pdf">k-comma numbers</a> [Cached copy, with permission]
%e A166513 The 4-digit terms 2676, 6246, 8349, 9427 occurring in A166512, can be split up in any of the 3 possible ways such that S(a,bcd), S(ab,cd), and S(abc,d) all contain abcd (concatenation, not product). Therefore they are in this sequence, and they are even hypercomma (or "phoenix") numbers (A166508).
%o A166513 (PARI) {for(n=1e4,1e5,/*is_A166513(n)=*/ my(c=3); for(d=1,#Str(n)-1, d+c>#Str(n) & break; 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 A166513 Cf. A166507, A166508, A166511, A166512.
%K A166513 base,nonn
%O A166513 1,1
%A A166513 _Eric Angelini_ and _M. F. Hasler_, Oct 29 2009