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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).
1
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
1,1
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)
LINKS
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.
EXAMPLE
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.
PROG
(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<q) => F[i]=F[i+n*k] & Aj (j<i) => ~(Ak (k<q) => F[j]=F[j+n*k])";
# Gandhar Joshi, Jan 25 2025
CROSSREFS
Cf. A003849, A339949 (length of the longest monochromatic arithmetic progression).
Sequence in context: A344546 A279719 A279672 * A089181 A028425 A224987
KEYWORD
nonn,changed
AUTHOR
Gandhar Joshi, Jul 31 2023
STATUS
approved