OFFSET
1,1
COMMENTS
Conjecture 1: Let v(0) = 2, v(1) = A, and v(n+1) = A*v(n) + v(n-1) for n > 0. Then A^2*det[v(j+k)+d(j,k)]_{1<=j,k<=n} = v(n+1)^2 - (A^2 + 4)*(n mod 2) for any positive integer n. In particular, a(n) = L(n+1)^2 - 5*(n mod 2) for all n > 0.
Conjecture 2: Let v(0) = 2, v(1) = A, and v(n+1) = A*v(n) - v(n-1) for n > 0. Then det[v(j+k)+d(j,k)]_{1<=j,k<=n} = u(n+1)^2 - n^2 for any positive integer n, where u(0) = 0, u(1) = 1, and u(n+1) = A*u(n) - u(n-1) for all n > 0.
Conjecture 3: Let F(n) denote the Fibonacci number A000045(n). Then, for any positive integer n, we have det[F(j+k) + d(j,k)]_{1<=j,k<=n} = F(n+1)^2 + (n mod 2).
LINKS
Han Wang and Zhi-Wei Sun, Evaluations of some Toeplitz-type determinants, arXiv:2206.12317 [math.NT], 2022.
EXAMPLE
a(2) = 16 since the determinant of the 2 X 2 matrix [L(1+1)+1, L(1+2); L(2+1), L(2+2)+1] = [4, 4; 4, 8] is 16.
MATHEMATICA
a[n_]:=a[n]=Det[Table[LucasL[j+k]+Boole[j==k], {j, 1, n}, {k, 1, n}]];
Table[a[n], {n, 1, 25}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 01 2023
STATUS
approved