A360278 Determinant of the matrix [L(j+k)+d(j,k)]_{1<=j,k<=n}, where L(n) denotes the Lucas number A000032(n), and d(j,k) is 1 or 0 according as j = k or not.

4, 16, 44, 121, 319, 841, 2204, 5776, 15124, 39601, 103679, 271441, 710644, 1860496, 4870844, 12752041, 33385279, 87403801, 228826124, 599074576, 1568397604, 4106118241, 10749957119, 28143753121, 73681302244, 192900153616, 505019158604, 1322157322201, 3461452807999, 9062201101801, 23725150497404, 62113250390416, 162614600673844, 425730551631121, 1114577054219519

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

Table of n, a(n) for n=1..35.

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

Cf. A000032, A000045.

Sequence in context: A289086 A018210 A054498 * A293629 A217553 A225379

Adjacent sequences: A360275 A360276 A360277 * A360279 A360280 A360281

Keyword

nonn

Author

Zhi-Wei Sun, Feb 01 2023

Status

approved