A392838 a(n) = (F(4*n + 3) - F(4*n - 1) + 5*n - 1)/5, where F(n) = A000045(n).

0, 3, 17, 107, 718, 4900, 33558, 229977, 1576247, 10803713, 74049700, 507544138, 3478759212, 23843770287, 163427632733, 1120149658775, 7677619978618, 52623190191472, 360684711361602, 2472169789339653, 16944503814015875, 116139356908771373, 796030994547383632, 5456077604922913942

Offset

0,2

Comments

a(n) is the trace of the n X n matrix whose generic element is given by F(2*i-1)*L(2*i) if i = j and F(2*j)*L(2*j-1) if i != j with 1 <= i,j <= n, where F(n) = A000045(n) and L(n) = A000032(n) (see Espinosa et al.).

Links

Table of n, a(n) for n=0..23.

Pedro Fernando Fernández Espinosa, Maritza Liliana Arciniegas Torres, and Camilo Andrés Acevedo Cadena, On Some Properties of Matrices with Entries Defined by Products of k-Fibonacci and k-Lucas Numbers, arXiv:2601.12644 [math.CO], 2026. See pp. 8-9.

Index entries for linear recurrences with constant coefficients, signature (9,-16,9,-1).

Formula

G.f.: x*(3 - 10*x + 2*x^2)/((1 - x)^2*(1 - 7*x + x^2)).

a(n) = 9*a(n-1) - 16*a(n-2) + 9*a(n-3) - a(n-4) for n > 3.

Example

a(3) = 107:
  [3, 12, 88]
  [1, 14, 88]
  [1, 12, 90]

Mathematica

a[n_]:=(Fibonacci[4n+3]-Fibonacci[4n-1]+5n-1)/5; Array[a, 24, 0]

Crossrefs

Cf. A000032, A000045, A033891, A172968.

Cf. A392837 (determinant), A392839 (eigenvalue with multiplicity 1), A392840 (permanent).

Sequence in context: A074563 A077145 A299039 * A123772 A212524 A103730

Adjacent sequences: A392835 A392836 A392837 * A392839 A392840 A392841

Keyword

nonn,easy

Author

Stefano Spezia, Jan 24 2026

Status

approved