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A364648 Starting position of the first occurrence of the longest monochromatic arithmetic progression of difference n in the Fibonacci infinite word (A003849).

%I A364648 #44 Feb 07 2025 16:08:51
%S A364648 2,3,20,16,11,20,0,143,2,11,54,8,32,2,11,7,70,3,7,0,986,10,3,7,16,11,
%T A364648 2,87,376,2,3,2,21,87,2,3,7,16,3,7,0,20,23,11,20,8,11,2,11,20,36,11,7,
%U A364648 0,6764,31,3,376,84,11,54,0,20,2,3,2,42,87,2,3,54,304
%N A364648 Starting position of the first occurrence of the longest monochromatic arithmetic progression of difference n in the Fibonacci infinite word (A003849).
%C A364648 From _Gandhar Joshi_, Jan 25 2025: (Start)
%C A364648 F(n) is the n-th Fibonacci number.
%C A364648 Conjecture: for n>0,
%C A364648 1. a(F(2n))=F(4n)-1; a(F(2n+1))=F(2n+3)-2.
%C A364648 2. a(F(6n)/2)=F(6n+3)/2-1; a(F(6n-3)/2)=F(6n)/2-2. (End)
%H A364648 Gandhar Joshi, <a href="/A364648/b364648.txt">Table of n, a(n) for n = 1..1973</a>
%H A364648 Ibai Aedo, U. Grimm, Y. Nagai, and P. Staynova, <a href="https://doi.org/10.1016/j.tcs.2022.08.013">Monochromatic arithmetic progressions in binary Thue-Morse-like words</a>, Theor. Comput. Sci., 934 (2022), 65-80.
%H A364648 Gandhar Joshi and D. Rust, <a href="https://arxiv.org/abs/2501.05830">Monochromatic arithmetic progressions in the Fibonacci word</a>, arXiv:2501.05830 [math.DS], 2025. See p.12.
%e A364648 For the difference n = 3, the longest monochromatic progression has length A339949(3) = 5 and thus defined by f(i)=f(i+3)=f(i+6)=f(i+9)=f(i+12), where f(i) is the i-th term of the Fibonacci word (A003849); the smallest i for which that holds is i=20, so a(3) = 20.
%o A364648 (Walnut)
%o A364648 # In the following line, replace every n with the desired constant difference, and every q with the longest MAP length for difference n given by A339949(n).
%o A364648 def f_n_map "?msd_fib Ak (k<q) => F[i]=F[i+n*k] & Aj (j<i) => ~(Ak (k<q) => F[j]=F[j+n*k])";
%o A364648 # _Gandhar Joshi_, Jan 25 2025
%Y A364648 Cf. A003849, A339949 (length of the longest monochromatic arithmetic progression).
%K A364648 nonn
%O A364648 1,1
%A A364648 _Gandhar Joshi_, Jul 31 2023