OFFSET
1,2
COMMENTS
From Robert Israel, May 19 2024: (Start)
Given a(1),...,a(n-1), the determinant of the Hankel matrix of [a(n-2*k), ..., a(n-1), x] is of the form A*x + B where A is the determinant of the Hankel matrix of [a(n-2*k), ..., a(n-2)]. Thus if A <> 0 there is only one x that makes this determinant 0. For a(n) there are at most n-1+ceil(n/2) "prohibited" values, namely a(1) to a(n-1) and ceil(n/2) values that make Hankel determinants 0. We conclude that a(n) always exists and a(n) <= 3*n/2. (End)
LINKS
Robert Israel, Table of n, a(n) for n = 1..750
Wikipedia, Hankel matrix
MAPLE
with(LinearAlgebra):
R:= [1]: S:= {1};
for i from 2 to 100 do
for y from 1 do
if member(y, S) then next fi;
found:= false;
for j from i-2 to 1 by -2 do if Determinant(HankelMatrix([op(R[j..i-1]), y]))=0 then found:= true; break fi od;
if not found then break fi;
od;
R:= [op(R), y];
S:= S union {y};
od:
R; # Robert Israel, May 19 2024
PROG
(Python)
from sympy import Matrix
from itertools import count
def A350348_list(nmax):
a=[]
for n in range(nmax):
a.append(next(k for k in count(1) if k not in a and 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))))
return a
CROSSREFS
KEYWORD
nonn
AUTHOR
Pontus von Brömssen, Dec 26 2021
STATUS
approved