A123183 a(1)=-1; a(2)=-1; a(3)=-2; a(n) = 4*a(n-1) - 3*a(n-2) for n >= 4.

-1, -1, -2, -5, -14, -41, -122, -365, -1094, -3281, -9842, -29525, -88574, -265721, -797162, -2391485, -7174454, -21523361, -64570082, -193710245, -581130734, -1743392201, -5230176602, -15690529805, -47071589414, -141214768241, -423644304722, -1270932914165, -3812798742494

Offset: 1

Author

Roger L. Bagula, Oct 02 2006

Keywords

• sign
• easy
• less

Comments

Essentially the same as A124302: a(n) = -A124302(n-1).

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Crossrefs

Cf. A124302.

Programs

Maple

a[1]:=-1: a[2]:=-1: a[3]:=-2: for n from 4 to 29 do a[n]:=4*a[n-1]-3*a[n-2] od: seq(a[n],n=1..29);

Mathematica

M = {{1, -1, 0}, {-1, 2, -1}, {0, -1, 1}} v[1] = {1, 0, 0} v[n_] := v[n] = M.v[n - 1] a1 = Table[ -v[n][[1]], {n, 1, 50}]

Formula

From Philippe Deléham, Dec 05 2008 [corrected by Sergei N. Gladkovskii, Dec 10 2012]: (Start)

a(n) = -A124302(n-1).

G.f.: -(1-3*x+x^2)/(1-4*x+3*x^2). (End)

G.f.: -1-x*G(0) where G(k) = 1 + 2*x/(1 - 2*x - x*(1-2*x)/(x + (1-2*x)*2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 10 2012

Extensions

Edited by N. J. A. Sloane, Oct 08 2006