cp's OEIS Frontend

A380593 Starting position of the first occurrence of the longest monochromatic arithmetic progression of difference n in the Rudin-Shapiro sequence (A020987).

%I A380593 #24 Feb 12 2025 21:40:11
%S A380593 7,14,28,28,31,43,95,56,43,62,453,86,99,190,39,112,495,86,366,124,81,
%T A380593 321,203,172,1006,81,233,380,2019,78,993,224,980,990,888,172,1084,732,
%U A380593 4057,248,2007,162,164,642,1215,406,1729,344,1398,2012,1988,162,1765
%N A380593 Starting position of the first occurrence of the longest monochromatic arithmetic progression of difference n in the Rudin-Shapiro sequence (A020987).
%C A380593 The length of this longest monochromatic progression is A364995(n).
%D A380593 B. Sobolewski, On monochromatic arithmetic progressions in binary words associated with pattern sequences, Theoretical Computer Science 1018 (2024), 114815.
%H A380593 Gandhar Joshi, <a href="/A380593/b380593.txt">Table of n, a(n) for n = 1..1022</a>
%H A380593 Bartosz Sobolewski, <a href="https://arxiv.org/abs/2204.05287">On monochromatic arithmetic progressions in binary words associated with pattern sequences</a>, arXiv:2204.05287 [math.CO], 2023.
%e A380593 For n = 3, the longest length of the monochromatic arithmetic progression in the Rudin-Shapiro sequence is given by A364995(3)=5. Let r(i) be the i-th term of the Rudin-Shapiro sequence (A020985). We have r(28) = r(31) = r(34) = r(37) = r(40), and no j<i exists such that r(j) = r(j+3) = r(j+2*3) = r(j+3*3) = r(j+4*3). So a(3)=28.
%o A380593 (Walnut)
%o A380593 # Replace in the code below: every n with the desired constant difference, and every q with the longest MAP length for difference n given by A364995(n).
%o A380593 def rs_n_map "Ak (k<q) => RS[i]=RS[i+n*k] & Aj (j<i) => ~(Ak (k<q) => RS[j]=RS[j+n*k])";
%o A380593 # While using Walnut to find values above and including a(1022), one has to run the code with constant value estimates and through trial and error hone in onto the smallest value that returns TRUE.
%Y A380593 Cf. A020987, A020985, A364995 (progression length).
%K A380593 nonn
%O A380593 1,1
%A A380593 _Gandhar Joshi_, Jan 27 2025