A165987 a(n) is the trace of the matrix f(X + n*f(X))/f(X), where X is the 2 X 2 matrix [-13, 17; 31, 97] and f(x) = x^3 - 5*x + 67.

1099258818702, 8792791182238, 29674231047422, 70337212371066, 137375369109982, 237382335220982, 376951744660878, 562677231386482, 801152429354606, 1098970972522062, 1462726494845662, 1899012630282218, 2414423012788542, 3015551276321446, 3708991054837742

Offset: 1

Author

A.K. Devaraj, Oct 03 2009

Keywords

• nonn
• easy
• less

Comments

Old name was: As mentioned in the short description (cf. A165806 & A165808) polynomials have the property: f(x + k*f(x)) is congruent to 0 mod(f(x)). This is true even if the variable is a square matrix. For this sequence let X be a 2x2 matrix (X belongs to Z): col1:-13, 31;col2: 17, 97. Let the polynomial be X^3 -5X + 67. The present sequence is a sequence of traces of the matrices resulting from the division of f(X + k*f(X))/f(X). Here k belongs to N.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A165806, A165808.

Programs

Maple

with(LinearAlgebra):
f:= x-> x^3-5*x+67:
a:= n-> (X-> Trace(f(X+n*f(X)).f(X)^(-1)))(<<-13|17>, <31|97>>):
seq(a(n), n=1..15);  # Alois P. Heinz, Mar 13 2024

Formula

G.f.: 2*(549309615337*x^3+2197877953721*x^2+549629409347*x+1)/(x-1)^4. - Alois P. Heinz, Mar 13 2024

Extensions

a(5)-a(15) added and edited by Alois P. Heinz, Mar 13 2024