OFFSET
1,2
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).
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
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 *)
PROG
(PARI) Vec(-x*(x^2-x-1)/(2*x^3-4*x^2-2*x+1) + O(x^100)) \\ Colin Barker, May 21 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Dec 21 2003
STATUS
approved