A350348 Lexicographically earliest sequence of distinct positive integers such that the Hankel matrix of any odd number of consecutive terms is invertible.

1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 12, 11, 13, 14, 16, 15, 17, 18, 19, 20, 22, 21, 23, 24, 25, 26, 29, 27, 28, 30, 31, 32, 33, 35, 34, 36, 37, 38, 39, 41, 40, 42, 44, 43, 45, 46, 47, 48, 50, 49, 51, 52, 53, 54, 56, 57, 55, 58, 59, 60, 61, 63, 62, 64, 65, 66, 67

Offset: 1

Author

Pontus von Brömssen, Dec 26 2021

Keywords

• nonn

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

Crossrefs

Cf. A350330, A350349, A350350.

Programs

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

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