A390505 - OEIS

%I #15 Dec 10 2025 15:45:49

%S 10,11,100,1011,1110,101,110,10011,101110,10101,11100,111011,10110,

%T 111,1000,11011,1001110,100101,1011100,10111011,1010110,10111,11110,

%U 1110011,11101110,1110101,101100,1011011,11110,1001,1010,100011,1101110,110101,10011100

%N a(n) = AN-run sequence of the n-th 01-word, where all 01-words are lexicographically ordered as in A076478; see Comments.

%C The AN-run (viz. Adjective-before-Noun) sequence of a 01-word (or binary representation, or vector of 0's and 1's) w, denoted by AN(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 AN(w) = [c(1)] d(1) [c(2)] d(2) ..., where [c(i)] is the base-2 representation of c(i). For NA-run sequences (Noun-before-Adjective), see A390508. For each w, the lengths of AN(w) and NA(w) are equal.

%e a(1) = AN(0) = 10 (one 0)

%e a(2) = AN(1) = 11 (one 1)

%e a(3) = AN(00) = 100 (two 0's)

%e a(8) = AN(001) = 10011 (two 0's and one 1)

%e For a(8), the "two" and "one" are adjectives (cardinals); the 0 and 1 are nouns (ordinals).

%t f[w_] := StringJoin[Map[IntegerString[Length[#], 2] <> ToString[#[[1]]] &, Split[Characters[w]]]];

%t s[n_] := Map[StringJoin[Map[ToString, Rest[IntegerDigits[#, 2]]]] &, Range[2, 2^(n + 1) - 1]];

%t Map[f, s[5]]

%t Map[# <> " -> " <> f[#] &, s[3]] // ColumnForm

%t (* _Peter J. C. Moses_, Nov 05 2025 *)

%Y Cf. A055186, A390506, A390507, A390508.

%K nonn,base

%O 1,1

%A _Clark Kimberling_, Dec 02 2025