A374139 a(n) is the determinant of the symmetric Toeplitz matrix of order n whose element (i,j) equals abs(i-j) or 1 if i = j.
1, 1, 0, -1, 1, 3, 0, -3, 1, 5, 0, -5, 1, 7, 0, -7, 1, 9, 0, -9, 1, 11, 0, -11, 1, 13, 0, -13, 1, 15, 0, -15, 1, 17, 0, -17, 1, 19, 0, -19, 1, 21, 0, -21, 1, 23, 0, -23, 1, 25, 0, -25, 1, 27, 0, -27, 1, 29, 0, -29, 1, 31, 0, -31, 1, 33, 0, -33, 1, 35, 0, -35, 1, 37, 0, -37
Offset: 0
Examples
a(4) = 1: [1, 1, 2, 3] [1, 1, 1, 2] [2, 1, 1, 1] [3, 2, 1, 1]
Links
- Max Alekseyev, Determinant of a certain Toeplitz matrix, MathOverflow, 2020.
- Index entries for linear recurrences with constant coefficients, signature (1,-2,2,-1,1).
Programs
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Mathematica
a[n_]:=Det[Table[If[i == j, 1, Abs[i - j]], {i, n}, {j, n}]]; Join[{1}, Array[a, 75]] -
PARI
a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, abs(i-j)))); \\ Michel Marcus, Jun 29 2024
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Python
from sympy import Matrix def A374139(n): return Matrix(n,n,[abs(j-k) if j!=k else 1 for j in range(n) for k in range(n)]).det() # Chai Wah Wu, Jul 01 2024
Formula
G.f.: (1 + x^2 - x^3 + x^4)/((1 - x)*(1 + x^2)^2).
a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) for n > 4.
E.g.f.: (exp(x) + (1 + x)*cos(x))/2.
For a proof of the generating function and the recursion formula, see MathOverflow link. - Sela Fried, Jul 09 2024
Comments