A316343 Concatenation of S[1], S[2], S[3], ..., where S[k] = 0 followed by m 1's, where m = (v+1)^2, v = exponent of highest power of 2 dividing k.

0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

0

Comments

A word that is morphic and recurrent, but neither pure morphic, uniform morphic, primitive morphic, nor uniformly recurrent.

Links

Jack W Grahl, Table of n, a(n) for n = 0..999

Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.

Example

S[4] = 0,1,1,1,1,1,1,1,1,1.

Mathematica

S[k_] := Join[{0}, Table[1, {(IntegerExponent[k, 2] + 1)^2}]];
Array[S, 20] // Flatten (* Jean-François Alcover, Nov 02 2018 *)

Prog

(Haskell)
a316343_count :: Integer -> Int
a316343_count n = fromIntegral ((v + 1) ^ 2) where
    v
        | n `mod` 2 == 1 = 0
        | otherwise      = 1 + (n `div` 2)
a316343_S :: Integer -> [Integer]
a316343_S n = 0 : (replicate (a316343_count n) 1)
a316343 :: [Integer]
a316343 = concatMap a316343_S [1..]
-- Jack W Grahl, Jul 23 2018

Crossrefs

Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.

Sequence in context: A277674 A309754 A112690 * A288864 A115971 A320007

Adjacent sequences: A316340 A316341 A316342 * A316344 A316345 A316346

Keyword

nonn

Author

N. J. A. Sloane, Jul 14 2018

Extensions

More terms from Jack W Grahl, Jul 23 2018

Status

approved