A316344 An example of a word that is uniform morphic, but neither pure morphic, primitive morphic, nor recurrent.

2, 2, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2

Offset: 0

Author

N. J. A. Sloane, Jul 14 2018

Keywords

• nonn
• easy

Links

• Zhuorui He, Table of n, a(n) for n = 0..10000 (first 1000 terms from Jack W Grahl)
• Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017. See Example 24.
• Jack W Grahl, Haskell code to generate this sequence

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.

Cf. A036577.

Programs

Mathematica

Join[{2, 2}, Differences[ThueMorse[Range[2, 100]]] + 1] (* Paolo Xausa, Jul 17 2025 *)

Formula

From Zhuorui He, Jul 11 2025: (Start)

a(n) = A010060(2*n+2) + A010060(max(2*n+1,4)).

a(n) = A036577(n+1) except a(1) = 2. (End)

Extensions

More terms from Jack W Grahl, Jul 23 2018