1,2

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 0-th, 1-st, and 2-nd 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).

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.

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

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

Adjacent sequences: A091137 A091138 A091139 this_sequence A091141 A091142 A091143

Sequence in context: A033139 A005354 A084084 this_sequence A052541 A024738 A052939

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2003