A191818 A sum-square avoiding sequence; fixed point of the map 0 -> 03; 1 -> 43; 3 -> 1; 4 -> 01.

0, 3, 1, 4, 3, 0, 1, 1, 0, 3, 4, 3, 4, 3, 0, 3, 1, 0, 1, 1, 0, 1, 1, 0, 3, 1, 4, 3, 0, 3, 4, 3, 4, 3, 0, 3, 4, 3, 4, 3, 0, 3, 1, 4, 3, 0, 1, 1, 0, 3, 1, 0, 1, 1, 0, 1, 1, 0, 3, 1, 0, 1, 1, 0, 1, 1, 0, 3, 1, 4, 3, 0, 1, 1, 0, 3, 4, 3, 4, 3, 0, 3, 1, 4, 3, 0, 3, 4, 3, 4, 3, 0, 3, 4, 3, 4, 3, 0, 3, 1

Offset

0,2

Comments

A "sum square" means two consecutive blocks of the same length and same sum.

This is a pure primitive morphic sequence. - N. J. A. Sloane, Jul 14 2018

References

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Links

David A. Corneth, Table of n, a(n) for n = 0..10000

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

J. Cassaigne, J. D. Currie, L. Schaeffer and J. Shallit, Avoiding Three Consecutive Blocks of the Same Size and Same Sum, arXiv:1106.5204 [cs.DM], 2011.

Michel Rigo, Relations on words, arXiv preprint arXiv:1602.03364 [cs.FL], 2016.

Formula

Fixed point of the map 0 -> 03; 1 -> 43; 3 -> 1; 4 -> 01.

Example

Start with 0 which maps to 03, then concatenate the image of the second digit, 3, which is 1, so we have 031. Then concatenate the image of the third digit, 1, which is 43. We then have 03143. Etc. - David A. Corneth, Aug 03 2017

Mathematica

Nest[Flatten[# /. {0 -> {0, 3}, 1 -> {4, 3}, 3 -> 1, 4 -> {0, 1}}] &, 0, 9] (* Michael De Vlieger, Aug 03 2017 *)

Prog

(PARI) first(n) = {my(res = [0, 3], i = 2, m = Map(Mat([0, [0, 3]; 1, [4, 3]; 3, [1]; 4, [0, 1]]))); while(#res < n, res = concat(res, mapget(m, res[i])); i++); res} \\ David A. Corneth, Aug 03 2017

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: A163359 A065256 A016573 * A055171 A337209 A101038

Adjacent sequences: A191815 A191816 A191817 * A191819 A191820 A191821

Keyword

nonn

Author

Jeffrey Shallit, Jun 28 2011

Status

approved