A174443 Generating function x/(1+4*x-8*x^2).

0, 1, -4, 24, -128, 704, -3840, 20992, -114688, 626688, -3424256, 18710528, -102236160, 558628864, -3052404736, 16678649856, -91133837312, 497964548096, -2720928890880, 14867431948288, -81237158920192, 443888091267072, -2425449636429824, 13252903275855872

Offset: 0

Author

Roger L. Bagula, Nov 28 2010

Keywords

• sign
• easy

Comments

Binet form: with a = 4/(1 + sqrt(3)); b = 4/(1 - sqrt(3)); we have a(n)=(a^n - b^n)/(a - b).

Alternative expressions for the roots are -2 +- 2*sqrt(3).

Limiting ratio a(n+1)/a(n) is:-5.464101615137755== -2-2*sqrt(3).

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A180222 (unsigned version of this sequence).

Programs

Mathematica

a = 4/(1 + Sqrt[3]); b = 4/(1 - Sqrt[3]);
f[n_] = (a^n - b^n)/(a - b);
Table[FullSimplify[f[n]], {n, 0, 30}]

Formula

a(0)=0, a(1)=1, a(n) = -4a(n-1) + 8a(n-2).

a(0)=0, a(n) = (-2)^(n-1)*A002605(n).

Extensions

Correction of the Mathematica code and better name by Joerg Arndt.

Formulae and edited by Ralf Stephan, Nov 27 2010