A329818 Look left and tell the least frequent digit string; a(0) = 0, a(n) = concat(c,s), where c > 0 is the number of times the string s, composed of one or more digits, has appeared in a(0) to a(n-1) such that a(n) is minimal. Digit strings with leading zeros are ignored.

0, 10, 11, 20, 12, 22, 30, 13, 23, 33, 40, 14, 24, 34, 44, 50, 15, 25, 35, 45, 55, 60, 16, 26, 36, 46, 56, 66, 70, 17, 27, 37, 47, 57, 67, 77, 80, 18, 28, 38, 48, 58, 68, 78, 88, 90, 19, 29, 39, 49, 59, 69, 79, 89, 99, 100, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 122

Offset: 0

Author

Scott R. Shannon, Nov 21 2019

Keywords

• nonn
• base

Comments

This is a variation on A329447. Instead of just considering the number of times the digits 0 to 9 have appeared so far in the sequence, we consider all numbers formed by the substrings of each previous entry; from single digits to the entire entry. Each value of a(n) is determined by considering the count of all the substrings occurring in a(0) to a(n-1) and then choosing the one which forms the lowest number when the count of that substring is concatenated with the substring itself.

The only restriction in the substring counting is that any substrings which have leading zeros are ignored. For example if a(n) was 2001, this would lead to incrementing the count of '0' by two, the count of '1' by one, the count of '2' by one, the count of '20' by 1, the count of '200' by 1, and the count of '2001' by 1. The substrings '00','01', and '001' are ignored and do not increment the count of '0' or '1'.

Unlike A329447, which is fairly constant in its growth, this sequence can have large decreases in its values from one entry to the next. These tend to be bunched and are followed by long series of entries with steady growth. For n up to 200000 the largest entry is 434397, the sequence decreases 4690 times, and the largest drop from one value to the next is 419882 which occurs at n = 199902.

This sequence is the same as A329447 up to a(55) = 100. After that, as a(2) = 11 and '11' has only appeared once, the next smallest value is that of 'one 11', that is 111.

Examples

			a(56) = 111 as a(2) = 11, and as 11 has only appeared once, the next smallest value is that of 'one 11', that is 111. After this entry the string '11' has now appeared three times, and the string '111' has appeared once.
a(199903) = 14342 as a(199902) = 434224 and as that contains the first appearance of the digit string '4342' in any entry, the next smallest value is that of 'one 4342', that is 14342. This leads to the largest drop in value for the first 200000 terms of the sequence. As a comparison after 199902 terms the digit '1' has occurred 153333 times.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 0..20000
• Eric Angelini, Look left and say, Nov 14 2019.
• Scott R. Shannon, Scatter plot for the first 200000 terms.

Crossrefs

Cf. A005150, A045918, A329447.