0,4

Definitions: (Start)

The k-th exterior power of a vector space V of dimension n is a vector subspace spanned by elements, called k-vectors, that are the exterior product of k vectors v_i in V.

Given a square matrix A that describes the vectors v_i in terms of a basis of V, the k-th exterior power of the matrix A is the matrix that represents the k-vectors in terms of the basis of V. (End)

Conjectures: (Start)

For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix A(n) defined as A[i,j,n] = n - abs((n + 1)/2 - j) - abs((n + 1)/2 - i) (see A349107). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of the matrix A(n), or the absolute value of the sum of all principal minors of A(n) of size 2.

For k > 2, the trace of the k-th exterior power of the matrix A(n) is equal to zero. (End)

The same conjectures hold for an n X n square matrix A(n) defined as A[i,j,n] = (n mod 2) + abs((n + 1)/2 - j) + abs((n + 1)/2 - i) (see A349108).

Winston de Greef, Table of n, a(n) for n = 0..10000

Wikipedia, Characteristic polynomial

Wikipedia, Exterior algebra

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

O.g.f.: 2*x^3*(1 + x^2)/((1 - x)^5*(1 + x)^3).

E.g.f.: (x*(x^3 + 6*x^2 + 3*x + 3)*cosh(x) + (x^4 + 6*x^3 + 9*x^2 - 3*x - 3)*sinh(x))/48.

a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 7.

a(n) = A338429(n-2)/2 for n > 2.

a(2*n-1) = 2*A006325(n).

a(2*n) = A112742(n).

Sum_{n>2} 1/a(n) = (45 - 2*Pi^2 - 4*sqrt(3)*Pi*tanh(sqrt(3)*Pi/2))/4 = 0.920755957767250147865...

Table[(2*n^4-6*(-1)^n*n^2-2*n^2+3*(-1)^n-3)/96, {n, 0, 45}]

(PARI) a(n) = (2*n^4 - 6*(-1)^n*n^2 - 2*n^2 + 3*(-1)^n - 3)/96 \\ Winston de Greef, Jan 28 2024

Cf. A000982 (trace of matrix A(n)), A317614 (elements sum of matrix A(n)), A349107, A349108.

Cf. A006325, A112742, A338429.

Sequence in context: A283290 A283379 A090889 * A096012 A261562 A102930

Adjacent sequences: A350047 A350048 A350049 * A350051 A350052 A350053

nonn,easy

Stefano Spezia, Dec 11 2021

approved