OFFSET
0,4
COMMENTS
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).
LINKS
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).
FORMULA
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...
MATHEMATICA
Table[(2*n^4-6*(-1)^n*n^2-2*n^2+3*(-1)^n-3)/96, {n, 0, 45}]
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Stefano Spezia, Dec 11 2021
STATUS
approved