A118702 a(n) = determinant of n X n circulant matrix whose first row is the first n Lucas numbers A000032, from L(0) to L(n-1).

1, 2, 3, 18, 0, 8347, -861952, 391524998, -359089453125, 893329160995712, -5499366235206395112, 87687141416511254851323, -3590079701896396800000000000, 381284797406693371431803926245802, -105147887074796935457211770823970013737, 75539385879999827508686234183177961972604928

Offset

0,2

Links

Table of n, a(n) for n=0..15.

Eric Weisstein's World of Mathematics, Circulant Matrix.

Example

a(4) = 0 because of the singular matrix:
  [2, 1, 3, 4]
  [4, 2, 1, 3]
  [3, 4, 2, 1]
  [1, 3, 4, 2].

Maple

L:= n-> (<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1]:
a:= n-> LinearAlgebra[Determinant](Matrix(n, shape=Circulant[[L(i)$i=0..n-1]])):
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 31 2026

Mathematica

circ[w_] := NestList[RotateRight, w, Length[w] - 1]; Table[ Det[ circ[ LucasL@ Range[0, n - 1]]], {n, 10}] (* Giovanni Resta, Jun 16 2016 *)

Crossrefs

A000032 Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2). A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000032, A048954, A052182, A066933, A086459, A086569.

Sequence in context: A132534 A263048 A390109 * A073524 A130226 A287621

Adjacent sequences: A118699 A118700 A118701 * A118703 A118704 A118705

Keyword

easy,sign

Author

Jonathan Vos Post, May 20 2006

Extensions

Corrected and extended by Giovanni Resta, Jun 16 2016

a(0)=1 prepended by Alois P. Heinz, Mar 31 2026

Status

approved