A098154 -id:A098154 - cp's OEIS frontend

A098153 Summarize the previous term in binary (in increasing order).

1, 11, 101, 10101, 100111, 1001001, 1000111, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001, 1101001

Offset: 1

Rick L. Shepherd, Aug 29 2004

Similar to A005151 but uses base 2: Let a(1)=1. Describing a(1) as "one 1" again gives a(2)=11 (same digit string as A005151 and similar sequences), but describing a(2) as "two 1's" gives a(3)=101 when the frequency of digit occurrence is written in binary and followed by the digit counted.

			Summarizing a(8) = 1101001 in increasing digit order, there are "three 0's, four 1's", so concatenating 11 0 100 1 gives a(9) = 1101001 (=a(10)=a(11)=...).

Cf. A098154 (ternary), A098155 (base 4), A005151 (decimal and digit strings for all other bases b >= 5).

a(n) = 1101001 for all n >= 8 (see example).

A098155 Summarize the previous term in base 4 (in increasing order).

Original entry on oeis.org

1, 11, 21, 1112, 3112, 211213, 312213, 212223, 1110213, 101011213, 201111213, 101112213, 101112213, 101112213, 101112213, 101112213, 101112213, 101112213, 101112213, 101112213, 101112213, 101112213, 101112213, 101112213, 101112213

Offset: 1

Rick L. Shepherd, Aug 29 2004

Let a(1)=1. Describing a(1) as "one 1" again gives a(2)=11 (same digit string as A005151 and similar sequences). Likewise, a(3) through a(8) have the same digit strings as the corresponding terms of A005151, but describing a(8) as "one 1, four 2s, one 3" gives a(9)=1110213 when the frequency of digit occurrence is written in base 4 and followed by the digit counted.

			Summarizing a(12) = 101112213 in increasing digit order, there are "one 0, five 1's, two 2s, one 3", so concatenating 1 0 11 1 2 2 1 3 gives a(13) = 101112213 (=a(14)=a(15)=...).

Onno M. Cain, Sela T. Enin, Inventory Loops (i.e. Counting Sequences) have Pre-period 2 max S_1 + 60, arXiv:2004.00209 [math.NT], 2020.

Cf. A098153 (binary), A098154 (ternary), A005151 (decimal and digit strings for all other bases b >= 5).

Nest[Append[#, FromDigits[Flatten@ Map[IntegerDigits[#, 4] & /@ Reverse@ # &, Tally@ Sort@ IntegerDigits@ #[[-1]] ] ]] &, {1}, 24] (* Michael De Vlieger, Jul 15 2020 *)

a(n) = 101112213 for all n >= 12 (see example).

cp's OEIS Frontend

A098153 Summarize the previous term in binary (in increasing order).

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