A171089 - OEIS

%I #23 Dec 26 2023 12:17:35

%S 6,4,16,34,96,244,646,1684,4416,11554,30256,79204,207366,542884,

%T 1421296,3720994,9741696,25504084,66770566,174807604,457652256,

%U 1198149154,3136795216,8212236484,21499914246,56287506244,147362604496,385800307234,1010038317216

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

%C 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

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

%H Vincenzo Librandi, <a href="/A171089/b171089.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).

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

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

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

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

%F a(n) = 2^(1-n)*((-2)^n+(3-sqrt(5))^n+(3+sqrt(5))^n). - _Colin Barker_, Oct 01 2016

%t f[n_] := 2 (LucasL@n^2 - (-1)^n); Array[f, 27, 0] (* _Robert G. Wilson v_, Sep 10 2010 *)

%t 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 *)

%o (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

%o (PARI) a(n) = round(2^(1-n)*((-2)^n+(3-sqrt(5))^n+(3+sqrt(5))^n)) \\ _Colin Barker_, Oct 01 2016

%o (PARI) Vec(2*(3-4*x-2*x^2)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ _Colin Barker_, Oct 01 2016

%Y Cf. A001254.

%K nonn,easy

%O 0,1

%A _R. J. Mathar_, Sep 08 2010

%E a(21) onwards from _Robert G. Wilson v_, Sep 10 2010