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]).

10, 20, 30, 40, 50, 60, 70, 80, 90, 109, 123, 148, 160, 163, 185, 198, 199, 241, 340, 362, 398, 422, 423, 444, 522, 540, 541, 550, 564, 597, 621, 622, 667, 683, 693, 724, 769, 770, 780, 806, 811, 835, 842, 888, 972, 1023, 1044, 1053, 1054, 1116, 1146, 1177

Offset: 1

Author

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

Keywords

• base
• nonn

Comments

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.

The sequence S is infinite and straightforward to compute, in contrast to the implicitly defined terms of A121805.

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.

Examples

			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.
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.
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).
		

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

See also subsequences A166508, containing A166509, and A166512, containing A166513.

Programs

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))}