A350351
Array read by antidiagonals: T(n,k) is the determinant of the Hankel matrix of the 2*n-1 consecutive terms of A350330 starting at the k-th term, n >= 0, k >= 1.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 2, -3, -4, 1, 1, 1, -4, -4, 1, 1, 1, -3, 3, -1, 1, 2, -2, -3, -3, -5, 13, 1, 2, -2, -5, 8, 14, -27, -40, 1, 1, 3, -3, 2, 4, 13, -22, -68, 1, 2, -3, -3, -1, 3, -3, 10, 48, 96, 1, 1, 1, -4, -4, 3, 3, 2, -48, 96, 80
Offset: 0
Array begins:
n\k| 1 2 3 4 5 6 7 8
---+--------------------------------
0 | 1 1 1 1 1 1 1 1
1 | 1 1 2 1 1 2 2 1
2 | 1 -3 1 1 -2 -2 3 -3
3 | -4 -4 -3 -3 -5 -3 -3 -4
4 | -4 3 -3 8 2 -1 -4 -4
5 | -1 -5 14 4 3 3 -1 7
6 | 13 -27 13 -3 3 -5 13 11
7 | -40 -22 10 2 -14 -32 -36 -46
8 | -68 48 -48 52 -40 16 8 27
A350364
Array read by antidiagonals: T(n,k) is the number of sequences of length n with terms in 1..k such that all Hankel matrices of an odd number of consecutive terms are invertible, n, k >= 0.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 0, 0, 1, 4, 9, 6, 0, 0, 1, 5, 16, 24, 10, 0, 0, 1, 6, 25, 58, 66, 14, 0, 0, 1, 7, 36, 118, 212, 174, 20, 0, 0, 1, 8, 49, 208, 560, 758, 462, 20, 0, 0, 1, 9, 64, 334, 1206, 2620, 2722, 1178, 22, 0, 0
Offset: 0
Array begins:
n\k| 0 1 2 3 4 5 6 7
---+-------------------------------------------
0 | 1 1 1 1 1 1 1 1
1 | 0 1 2 3 4 5 6 7
2 | 0 1 4 9 16 25 36 49
3 | 0 0 6 24 58 118 208 334
4 | 0 0 10 66 212 560 1206 2282
5 | 0 0 14 174 758 2620 6932 15506
6 | 0 0 20 462 2722 12277 39871 105405
7 | 0 0 20 1178 9628 57084 228451 714878
8 | 0 0 22 3036 34132 265659 1309476 4849364
A350348
Lexicographically earliest sequence of distinct positive integers such that the Hankel matrix of any odd number of consecutive terms is invertible.
Original entry on oeis.org
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
-
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
-
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
A350349
Lexicographically earliest increasing sequence of positive integers such that the Hankel matrix of any odd number of consecutive terms is invertible.
Original entry on oeis.org
1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 16, 17, 19, 20, 21, 22, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 38, 39, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 80, 81, 82, 83, 85, 86, 87, 88, 90
Offset: 1
-
from sympy import Matrix
from itertools import count
def A350349_list(nmax):
a=[1]
for n in range(1,nmax):
a.append(next(k for k in count(a[-1]+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))))
return a
A350350
Lexicographically earliest nondecreasing sequence of positive integers such that the Hankel matrix of any odd number of consecutive terms is invertible.
Original entry on oeis.org
1, 1, 2, 2, 3, 3, 5, 5, 6, 6, 7, 7, 9, 9, 11, 11, 12, 12, 14, 14, 15, 15, 16, 16, 18, 18, 19, 19, 21, 21, 24, 24, 25, 25, 26, 26, 28, 28, 29, 29, 30, 30, 32, 32, 34, 34, 35, 35, 37, 37, 38, 38, 39, 39, 41, 41, 43, 43, 46, 46, 48, 48, 50, 50, 51, 51, 52, 52, 55
Offset: 1
-
from sympy import Matrix
from itertools import count
def A350350_list(nmax):
a=[1]
for n in range(1,nmax):
a.append(next(k for k in count(a[-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))))
return a
A350365
Array read by antidiagonals: T(n,k) is the number of sequences of length 2*n+1 with terms in 0..k such that the Hankel matrix of the sequence is singular, but the Hankel matrix of any proper subsequence with an odd number of consecutive terms is invertible, n, k >= 0.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 6, 6, 10, 0, 0, 1, 7, 16, 52, 0, 0, 0, 1, 8, 36, 148, 116, 8, 0, 0, 1, 9, 58, 448, 644, 528, 12, 0, 0, 1, 12, 82, 885, 2932, 4032, 1326, 0, 0, 0
Offset: 0
Array begins:
n\k| 0 1 2 3 4 5
---+----------------------
0 | 1 1 1 1 1 1
1 | 0 1 2 3 6 7
2 | 0 0 2 6 16 36
3 | 0 0 10 52 148 448
4 | 0 0 0 116 644 2932
For n = 2 and k = 4, the following T(2,4) = 16 sequences are counted:
(1, 1, 2, 2, 4),
(1, 2, 1, 2, 1),
(1, 2, 2, 4, 4),
(1, 3, 1, 3, 1),
(1, 4, 1, 4, 1),
(2, 1, 2, 1, 2),
(2, 3, 2, 3, 2),
(2, 4, 2, 4, 2),
(3, 1, 3, 1, 3),
(3, 2, 3, 2, 3),
(3, 4, 3, 4, 3),
(4, 1, 4, 1, 4),
(4, 2, 2, 1, 1),
(4, 2, 4, 2, 4),
(4, 3, 4, 3, 4),
(4, 4, 2, 2, 1).
Showing 1-6 of 6 results.
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