A307887 -id:A307887 - cp's OEIS frontend

A350954 Maximal determinant of an n X n symmetric Toeplitz matrix using the integers 1 to n.

1, 1, 3, 15, 100, 3091, 49375, 1479104, 43413488, 1539619328, 64563673460, 2877312739624, 252631974548628

Offset: 0

Stefano Spezia, Jan 27 2022

			a(3) = 15:
    1    3    2
    3    1    3
    2    3    1
a(4) = 100:
    2    1    4    3
    1    2    1    4
    4    1    2    1
    3    4    1    2
a(5) = 3091:
    3    5    1    2    4
    5    3    5    1    2
    1    5    3    5    1
    2    1    5    3    5
    4    2    1    5    3

Lucas A. Brown, A350953+4+A356865.py
Wikipedia, Toeplitz Matrix

Cf. A307887, A350931, A350953 (minimal), A356865 (minimal nonzero absolute value).

from itertools import permutations
from sympy import Matrix
def A350954(n): return max(Matrix([p[i:0:-1]+p[0:n-i] for i in range(n)]).det() for p in permutations(range(1,n+1))) # Chai Wah Wu, Jan 27 2022

a(9) from Alois P. Heinz, Jan 27 2022

a(10)-a(12) from Lucas A. Brown, Sep 01 2022

A350953 Minimal determinant of an n X n symmetric Toeplitz matrix using the integers 1 to n.

Original entry on oeis.org

1, 1, -3, -12, -100, -1749, -47600, -800681, -39453535, -1351201968, -66984136299, -2938096403400, -235011452211680

Offset: 0

Stefano Spezia, Jan 27 2022

			a(3) = -12:
    2    3    1
    3    2    3
    1    3    2
a(4) = -100:
    3    4    1    2
    4    3    4    1
    1    4    3    4
    2    1    4    3
a(5) = -1749:
    5    4    1    3    2
    4    5    4    1    3
    1    4    5    4    1
    3    1    4    5    4
    2    3    1    4    5

Lucas A. Brown, A350953+4+A356865.py
Wikipedia, Toeplitz Matrix

Cf. A307887, A350930, A350954 (maximal), A356865 (minimal nonzero absolute value).

from itertools import permutations
from sympy import Matrix
def A350953(n): return min(Matrix([p[i:0:-1]+p[0:n-i] for i in range(n)]).det() for p in permutations(range(1,n+1))) # Chai Wah Wu, Jan 27 2022

a(9) from Alois P. Heinz, Jan 27 2022

a(10)-a(12) from Lucas A. Brown, Sep 01 2022

A308110 Least number k such that the determinant of the symmetric Hankel matrix formed by its decimal digits is equal to n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 302, 65, 42, 76, 10320, 41, 40, 98, 522, 413, 64, 52, 354, 645, 51, 50, 142, 63, 86, 1534, 13112, 1387, 62, 74, 514, 61, 60, 635, 978, 85, 73, 1431, 502, 2677, 152, 72, 746, 625, 71, 70, 378, 2415, 254, 475, 366, 83, 95, 263, 33442

Offset: 0

Paolo P. Lava, May 13 2019

The first nontrivial values for which a(n) = n are at n = 288 and n = 26825.
When a(n) < n: 168, 182, 232, 234, 252, 272, 280, 300, 304, 320, 324, 325, etc. - Robert G. Wilson v, May 14 2019
Records: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 302, 10320, 13112, 33442, 53242, 55262, 58493, 74959, 1021310, 1124232, 1272626, 1400230, 2034050, 2514162, 3043724, 4986388, 5604351, 106071534, 108162262, 117200232, 128580276, 134314966, 163332550, 165244716, 166811088, 225231732, 229330425, etc. - Robert G. Wilson v, May 15 2019

			                        | 3 0 2 |
a(10) = 302 because det | 0 2 0 | = 10.
                        | 2 0 3 |
.
                         | 1 0 3 2 0 |
                         | 0 3 2 0 2 |
a(14)= 10320 because det | 3 2 0 2 3 | = 14.
                         | 2 0 2 3 0 |
                         | 0 2 3 0 1 |

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
Wikipedia, Hankel matrix

Cf. A306593, A307887, A308126.

with(numtheory): with(linalg): P:=proc(q) local c,d,i,k,n,t: print(0);
for i from 1 to q do for n from 1 to q do c:=convert(n, base, 10): t:=[]:
for k from 1 to nops(c) do t:=[op(t),0]: od: d:=t: t:=[]:
for k from 1 to nops(c) do t:=[op(t),d]: t[k,-k]:=1: od:
if det(evalm(toeplitz(c) &* t))=i then print(n); break: fi:
od: od: end: P(10^8);

f[n_] := Block[{k = 0}, While[id = IntegerDigits@ k; Det[HankelMatrix[id, Reverse@ id]] != n, k++]; k]; Array[f, 60, 0] (* Robert G. Wilson v, May 14 2019 *)

Offset corrected by Robert G. Wilson v, May 14 2019

cp's OEIS Frontend

A350954 Maximal determinant of an n X n symmetric Toeplitz matrix using the integers 1 to n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Extensions

A350953 Minimal determinant of an n X n symmetric Toeplitz matrix using the integers 1 to n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Extensions

A308110 Least number k such that the determinant of the symmetric Hankel matrix formed by its decimal digits is equal to n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions