A390505 - OEIS

A390505

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

10, 11, 100, 1011, 1110, 101, 110, 10011, 101110, 10101, 11100, 111011, 10110, 111, 1000, 11011, 1001110, 100101, 1011100, 10111011, 1010110, 10111, 11110, 1110011, 11101110, 1110101, 101100, 1011011, 11110, 1001, 1010, 100011, 1101110, 110101, 10011100

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OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..35.

EXAMPLE

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

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

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

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

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

MATHEMATICA

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

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

Map[f, s[5]]