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

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

Offset: 1

Author

Gandhar Joshi, Jul 31 2023

Keywords

• nonn

Comments

From Gandhar Joshi, Jan 25 2025: (Start)

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

Conjecture: for n>0,

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

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

Examples

			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.
		

Links

• Gandhar Joshi, Table of n, a(n) for n = 1..1973
• Ibai Aedo, U. Grimm, Y. Nagai, and P. Staynova, Monochromatic arithmetic progressions in binary Thue-Morse-like words, Theor. Comput. Sci., 934 (2022), 65-80.
• Gandhar Joshi and D. Rust, Monochromatic arithmetic progressions in the Fibonacci word, arXiv:2501.05830 [math.DS], 2025. See p.12.

Crossrefs

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

Programs

Walnut

# 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).
def f_n_map "?msd_fib Ak (k F[i]=F[i+n*k] & Aj (j ~(Ak (k F[j]=F[j+n*k])";
# Gandhar Joshi, Jan 25 2025