A364648 - OEIS

%I #44 Feb 07 2025 16:08:51

%S 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 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 0,6764,31,3,376,84,11,54,0,20,2,3,2,42,87,2,3,54,304

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

%C From _Gandhar Joshi_, Jan 25 2025: (Start)

%C F(n) is the n-th Fibonacci number.

%C Conjecture: for n>0,

%C 1. a(F(2n))=F(4n)-1; a(F(2n+1))=F(2n+3)-2.

%C 2. a(F(6n)/2)=F(6n+3)/2-1; a(F(6n-3)/2)=F(6n)/2-2. (End)

%H Gandhar Joshi, <a href="/A364648/b364648.txt">Table of n, a(n) for n = 1..1973</a>

%H 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 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 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 (Walnut)

%o # 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 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 # _Gandhar Joshi_, Jan 25 2025

%Y Cf. A003849, A339949 (length of the longest monochromatic arithmetic progression).

%K nonn,changed

%O 1,1

%A _Gandhar Joshi_, Jul 31 2023