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A sum-square avoiding sequence; fixed point of the map 0 -> 03; 1 -> 43; 3 -> 1; 4 -> 01.
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%I #45 Jul 15 2018 00:21:31

%S 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,

%T 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,

%U 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

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

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

%C This is a pure primitive morphic sequence. - _N. J. A. Sloane_, Jul 14 2018

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

%H David A. Corneth, <a href="/A191818/b191818.txt">Table of n, a(n) for n = 0..10000</a>

%H Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, <a href="https://arxiv.org/pdf/1711.10807.pdf">A Taxonomy of Morphic Sequences</a>, arXiv preprint arXiv:1711.10807, Nov 29 2017

%H J. Cassaigne, J. D. Currie, L. Schaeffer and J. Shallit, <a href="http://arxiv.org/abs/1106.5204">Avoiding Three Consecutive Blocks of the Same Size and Same Sum</a>, arXiv:1106.5204 [cs.DM], 2011.

%H Michel Rigo, <a href="http://arxiv.org/abs/1602.03364">Relations on words</a>, arXiv preprint arXiv:1602.03364 [cs.FL], 2016.

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

%e 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

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

%o (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

%Y 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.

%K nonn

%O 0,2

%A _Jeffrey Shallit_, Jun 28 2011