This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350330 #25 May 22 2024 11:32:53 %S A350330 1,1,2,1,1,2,2,1,2,1,1,2,1,2,3,1,1,2,1,1,2,2,1,2,1,1,2,2,3,2,2,1,1,2, %T A350330 2,3,1,1,2,1,1,2,2,1,2,1,1,2,1,2,3,1,1,2,1,1,2,2,1,2,1,1,2,2,3,2,2,1, %U A350330 1,2,3,1,1,2,1,1,2,2,1,2,1,1,2,1,2,3,1 %N A350330 Lexicographically earliest sequence of positive integers such that the Hankel matrix of any odd number of consecutive terms is invertible. %C A350330 No linear relation of the form c_1*a(j) + ... + c_k*a(j+k-1) = 0, with at least one c_i nonzero, holds for k consecutive values of j. %C A350330 Is a(n) <= 3 for all n? (It is true for n <= 400.) If not, what is the largest term? Or is the sequence unbounded? %C A350330 There seems to be some regularity in the sequence of values of n for which a(n) > 2: 15, 29, 36, 51, 65, 71, 86, 100, ... . The first differences of these are: 14, 7, 15, 14, 6, 15, 14, 5, 15, 14, 3, 15, 14, 1, 15, 13, 11, 15, 14, 7, 15, 14, 5, 15, 14, 3, 15, 14, 1, ... . The differences are all less than or equal to 15, because A350364(15,2) = 0. %C A350330 Agrees with A154402 for the first 20 terms, but differs on the 21st. %H A350330 Robert Israel, <a href="/A350330/b350330.txt">Table of n, a(n) for n = 1..1000</a> (1..400 from _Pontus von Brömssen) %H A350330 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hankel_matrix">Hankel matrix</a> %e A350330 a(15) = 3, because the Hankel matrix of (a(11), ..., a(15)) is %e A350330 [1 2 1 ] %e A350330 [2 1 2 ] %e A350330 [1 2 a(15)], %e A350330 which is singular if a(15) = 1, and the Hankel matrix of (a(5), ..., a(15)) is %e A350330 [1 2 2 1 2 1 ] %e A350330 [2 2 1 2 1 1 ] %e A350330 [2 1 2 1 1 2 ] %e A350330 [1 2 1 1 2 1 ] %e A350330 [2 1 1 2 1 2 ] %e A350330 [1 1 2 1 2 a(15)], %e A350330 which is singular if a(15) = 2, but if a(15) = 3 the Hankel matrix of (a(k), ..., a(15)) is invertible for all odd k <= 15. %o A350330 (Python) %o A350330 from sympy import Matrix %o A350330 from itertools import count %o A350330 def A350330_list(nmax): %o A350330 a=[] %o A350330 for n in range(nmax): %o A350330 a.append(next(k for k in count(1) if all(Matrix((n-r)//2+1,(n-r)//2+1,lambda i,j:(a[r:]+[k])[i+j]).det()!=0 for r in range(n-2,-1,-2)))) %o A350330 return a %o A350330 (Python) %o A350330 # Faster version using numpy instead of sympy. %o A350330 # Due to floating point errors, the results may be inaccurate for large n. Correctness verified up to n=400 for numpy 1.20.2. %o A350330 from numpy import array %o A350330 from numpy.linalg import det %o A350330 from itertools import count %o A350330 def A350330_list(nmax): %o A350330 a=[] %o A350330 for n in range(nmax): %o A350330 a.append(next(k for k in count(1) if all(abs(det(array([[(a[r:]+[k])[i+j] for j in range((n-r)//2+1)] for i in range((n-r)//2+1)])))>0.5 for r in range(n-2,-1,-2)))) %o A350330 return a %Y A350330 Cf. A350351, A350348, A350349, A350350, A350364, A154402. %K A350330 nonn %O A350330 1,3 %A A350330 _Pontus von Brömssen_, Dec 25 2021