This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350050 #27 Jan 28 2024 18:10:31
%S A350050 0,0,0,2,4,14,24,52,80,140,200,310,420,602,784,1064,1344,1752,2160,
%T A350050 2730,3300,4070,4840,5852,6864,8164,9464,11102,12740,14770,16800,
%U A350050 19280,21760,24752,27744,31314,34884,39102,43320,48260,53200,58940,64680,71302,77924,85514
%N A350050 a(n) = (2*n^4 - 6*(-1)^n*n^2 - 2*n^2 + 3*(-1)^n - 3)/96.
%C A350050 Definitions: (Start)
%C A350050 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.
%C A350050 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)
%C A350050 Conjectures: (Start)
%C A350050 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.
%C A350050 For k > 2, the trace of the k-th exterior power of the matrix A(n) is equal to zero. (End)
%C A350050 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).
%H A350050 Winston de Greef, <a href="/A350050/b350050.txt">Table of n, a(n) for n = 0..10000</a>
%H A350050 Wikipedia, <a href="https://en.wikipedia.org/wiki/Characteristic_polynomial">Characteristic polynomial</a>
%H A350050 Wikipedia, <a href="https://en.wikipedia.org/wiki/Exterior_algebra">Exterior algebra</a>
%H A350050 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A350050 O.g.f.: 2*x^3*(1 + x^2)/((1 - x)^5*(1 + x)^3).
%F A350050 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.
%F A350050 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.
%F A350050 a(n) = A338429(n-2)/2 for n > 2.
%F A350050 a(2*n-1) = 2*A006325(n).
%F A350050 a(2*n) = A112742(n).
%F A350050 Sum_{n>2} 1/a(n) = (45 - 2*Pi^2 - 4*sqrt(3)*Pi*tanh(sqrt(3)*Pi/2))/4 = 0.920755957767250147865...
%t A350050 Table[(2*n^4-6*(-1)^n*n^2-2*n^2+3*(-1)^n-3)/96,{n,0,45}]
%o A350050 (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
%Y A350050 Cf. A000982 (trace of matrix A(n)), A317614 (elements sum of matrix A(n)), A349107, A349108.
%Y A350050 Cf. A006325, A112742, A338429.
%K A350050 nonn,easy
%O A350050 0,4
%A A350050 _Stefano Spezia_, Dec 11 2021