A091140 a(n) = 2*a(n-1) + 4*a(n-2) - 2*a(n-3) with initial terms 1, 3, 9.

1, 3, 9, 28, 86, 266, 820, 2532, 7812, 24112, 74408, 229640, 708688, 2187120, 6749712, 20830528, 64285664, 198394016, 612269632, 1889544000, 5831378496, 17996393728, 55539213440, 171401244800, 528966555904, 1632459664128, 5037983062272, 15547871669248

Offset: 1

Author

Gary W. Adamson, Dec 21 2003

Keywords

• nonn
• easy

Comments

One of 3 related sequences generated from finite difference operations. Let r(1)=s(1)=t(1)=1. Given r(n), s(n) and t(n), let f(x) = r(n) x^2 + s(n) x + t(n) and let r(n+1), s(n+1) and t(n+1) be the 0th, 1st and 2nd differences of f(x) at x=1. I.e., r(n+1) = f(1) = r(n)+s(n)+t(n), s(n+1) = f(2)-f(1) = 3r(n)+s(n) and t(n+1) = f(3)-2f(2)+f(1) = 2r(n). This sequence gives r(n).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,4,-2).

Crossrefs

Cf. s(n) = A091141(n), t(n) = A091142(n).

Programs

Mathematica

a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {2, 0, 0}}, n-1].{{1}, {1}, {1}})[[1, 1]]
LinearRecurrence[{2,4,-2},{1,3,9},30] (* Harvey P. Dale, May 18 2021 *)

PARI

Vec(-x*(x^2-x-1)/(2*x^3-4*x^2-2*x+1) + O(x^100)) \\ Colin Barker, May 21 2015

Formula

Let v(n) be the column vector with elements r(n), s(n), t(n); then v(n) = [1 1 1 / 3 1 0 / 2 0 0] v(n-1).

The limit as n->infinity of a(n+1)/a(n) is the largest root of x^3 - 2x^2 - 4x + 2 = 0, which is about 3.086130197651494.

G.f.: -x*(x^2-x-1) / (2*x^3-4*x^2-2*x+1). - Colin Barker, May 21 2015