A171089 a(n) = 2*(Lucas(n)^2 - (-1)^n).

6, 4, 16, 34, 96, 244, 646, 1684, 4416, 11554, 30256, 79204, 207366, 542884, 1421296, 3720994, 9741696, 25504084, 66770566, 174807604, 457652256, 1198149154, 3136795216, 8212236484, 21499914246, 56287506244, 147362604496, 385800307234, 1010038317216

Offset: 0

Author

R. J. Mathar, Sep 08 2010

Keywords

• nonn
• easy

Comments

In Thomas Koshy's book on Fibonacci and Lucas numbers, the formula for even-indexed Lucas numbers in terms of squares of Lucas numbers (A001254) is erroneously given as L(2n) = 2L(n)^2 + 2(-1)^(n - 1) on page 404 as Identity 34.7. - Alonso del Arte, Sep 07 2010

References

• Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

Cf. A001254.

Programs

Magma

I:=[6, 4, 16]; [n le 3 select I[n] else 2*Self(n-1) + 2*Self(n-2) - Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

Mathematica

f[n_] := 2 (LucasL@n^2 - (-1)^n); Array[f, 27, 0] (* Robert G. Wilson v, Sep 10 2010 *)
CoefficientList[Series[2*(3 - 4*x - 2*x^2)/((1 + x)*(x^2 - 3*x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)

PARI

a(n) = round(2^(1-n)*((-2)^n+(3-sqrt(5))^n+(3+sqrt(5))^n)) \\ Colin Barker, Oct 01 2016

PARI

Vec(2*(3-4*x-2*x^2)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Oct 01 2016

Formula

a(n) = 2*(A000032(n))^2 -2*(-1)^n.

a(n) = 2*A047946(n).

a(n) = 2*a(n-1) + 2*a(n-2) -a(n-3).

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

a(n) = 2^(1-n)*((-2)^n+(3-sqrt(5))^n+(3+sqrt(5))^n). - Colin Barker, Oct 01 2016

Extensions

a(21) onwards from Robert G. Wilson v, Sep 10 2010