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

1,2

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

Cf. A350330, A350349, A350350.

Sequence in context: A129607 A130340 A130339 * A058812 A320454 A320455

Adjacent sequences: A350345 A350346 A350347 * A350349 A350350 A350351

Keyword

nonn

Author

Pontus von Brömssen, Dec 26 2021

Status

approved