A091141 a(n) = 2*a(n-1) + 4*a(n-2) - 2*a(n-3) with initial terms 1, 4, 13.

1, 4, 13, 40, 124, 382, 1180, 3640, 11236, 34672, 107008, 330232, 1019152, 3145216, 9706576, 29955712, 92447296, 285304288, 880486336, 2717295232, 8385927232, 25880062720, 79869243904, 246486884224, 760690618624, 2347590286336, 7244969278720, 22358918465536

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

Programs

Mathematica

a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {2, 0, 0}}, n-1].{{1}, {1}, {1}})[[2, 1]]
LinearRecurrence[{2,4,-2},{1,4,13},30] (* Harvey P. Dale, Jun 19 2018 *)

PARI

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