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]).

1023, 1044, 1521, 1657, 1789, 1984, 2191, 2263, 2451, 2466, 2523, 2676, 2783, 2824, 3066, 3268, 3589, 3602, 3631, 4051, 4113, 4149, 4159, 4213, 4315, 4611, 4685, 4781, 4969, 5133, 5526, 6053, 6165, 6246, 6445, 6650, 6712, 6893, 7350, 7668, 8011, 8144

Offset: 1

Author

Eric Angelini and M. F. Hasler, Oct 28 2009

Keywords

• base
• easy
• nonn

Comments

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]).

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.

Since the 4-digit terms remind of year numbers, the terminology of bicommatile (in analogy with bissextile) years has been introduced (as a joke).

Examples

			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).
Therefore the smallest term in this sequence is a(1)=1023, which occurs in the sequences S(102,3) and S(10,23).
		

Links

• E. Angelini, Comma numbers, SeqFan mailing list, Oct 15 2009
• E. Angelini, k-comma numbers, Oct. 2009.
• E. Angelini, k-comma numbers [Cached copy, with permission]

Crossrefs

Cf. A166507, A166508, A166511, A166513.

Programs

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))}