A390508 a(n) = index of the NA-run sequence of the n-th 01-word, where all 01-words are lexicographically ordered (i.e., enumerated) as in A076478; see Comments.

4, 6, 9, 22, 28, 13, 10, 42, 92, 45, 57, 118, 56, 14, 19, 46, 172, 85, 185, 374, 184, 46, 58, 234, 476, 237, 113, 230, 60, 27, 20, 82, 188, 93, 345, 694, 344, 86, 186, 746, 1500, 749, 369, 742, 188, 91, 115, 238, 940, 469, 953, 1910, 952, 238, 114, 458, 924

Offset

1,1

Comments

The NA-run sequence of a 01-word (or binary representation, or vector of 0's and 1's), w, denoted by NA(w), is defined as follows: represent w as c(1) d(1) c(2) d(2) ..., where c(i) is the number of digits d(i) in the i-th run in w. Then NA(w) = d(1) [c(1)] d(2) [c(2)] ..., where [c(i)] is the binary representation of c(i). For AN-run sequences, see A390505. For each w, the lengths of AN(w) and NA(w) are equal.

Links

Table of n, a(n) for n=1..57.

Example

The first seven 01-words are 0, 1, 00, 01, 10, 11, 000. Starting with the first word, 0, we have c(1) = 1 and d(1) = 0, so that NA(0) = 01, which is the fourth 01-word in the sequence (0,1,00,01,10,11,000,...). The NA-run sequence of the n-th 01 word, for n>=1, is (01, 11, 010, 0111, 1101, 110, 011, 01011, 011101, 01110, 11010, 110111, 11001, 111, 0100, 01111, 0101101, 010110, 0111010, 01110111, 0111001, 01111, 11011, 1101011, 11011101, 1101110, 110010, 1100111, 11101, 1100, ...). Using the lexicographic ordering of all 01 words as in A076478, the indices of the words 01, 11, 010, ... are 4, 6, 9, ... .
a(1) = 4, which is the index of (NA(0) = 01) (0 one time; i.e., the noun is 0, and the adjective is 1.)
a(2) = 6, the index of (NA(1) = 11) (1 one time)
a(3) = 9, the index of (NA(00) = 010) (0 two times)
a(8) = 42, the index of (NA(001) = 01011) (0 twice and 1 once; i.e. 0 and 1 are nouns (ordinals), and "twice" and "once" are adjectives (cardinals)).

Mathematica

Map[2^Length[#] - 2 + FromDigits[#, 2] + 1 &,
 Map[Flatten[{First[#], IntegerDigits[Length[#], 2]} & /@ Split[#]] &,
   Flatten[Map[Tuples[{0, 1}, {#}] &, Range[6]], 1]]]
(* Peter J. C. Moses, Dec 17 2025 *)

Crossrefs

Cf. A055168, A390505, A390509, A390510.

Sequence in context: A115681 A257842 A239307 * A137253 A035135 A046338

Adjacent sequences: A390505 A390506 A390507 * A390509 A390510 A390511

Keyword

nonn

Author

Clark Kimberling, Dec 04 2025

Status

approved