A091142 a(n) = 2*a(n-1) + 4*a(n-2) - 2*a(n-3) with initial terms 1, 2, 6.

1, 2, 6, 18, 56, 172, 532, 1640, 5064, 15624, 48224, 148816, 459280, 1417376, 4374240, 13499424, 41661056, 128571328, 396788032, 1224539264, 3779088000, 11662756992, 35992787456, 111078426880, 342802489600, 1057933111808, 3264919328256, 10075966124544

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 t(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. r(n) = A091140(n), s(n) = A091141(n).

Programs

Mathematica

a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {2, 0, 0}}, n-1].{{1}, {1}, {1}})[[3, 1]]

PARI

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