|
%I #19 Sep 08 2022 08:46:08
%S 4,0,18,48,294,1210,5508,23548,101614,433200,1845738,7840998,33282564,
%T 141149320,598366458,2535856048,10745092894,45524786370,192866785668,
%U 817050731748,3461224027254,14662350247600,62111682111618,263111844646798,1114566304573444
%N a(n) = L(n)^3 - L(n)^2, where L(n) is the n-th Lucas number (A000032).
%H Colin Barker, <a href="/A244310/b244310.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,2,-22,-4,14,-1,-1).
%F G.f.: 2*(x^6-8*x^5+17*x^4+23*x^3+5*x^2-10*x+2) / ((x+1)*(x^2-3*x+1)*(x^2-x-1)*(x^2+4*x-1)).
%F a(n) = A045991(A000032(n)). - _Michel Marcus_, Jun 25 2014
%e a(3) is 48 because L(3)^3 - L(3)^2 = 4^3 - 4^2 = 48.
%t CoefficientList[Series[2 (x^6 - 8 x^5 + 17 x^4 + 23 x^3 + 5 x^2 - 10 x + 2)/((x + 1) (x^2 - 3 x + 1) (x^2 - x - 1) (x^2 + 4 x - 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 26 2014 *)
%t Table[LucasL[n]^3 - LucasL[n]^2, {n,0,50}] (* _G. C. Greubel_, Oct 13 2018 *)
%o (PARI)
%o lucas(n) = if(n==0, 2, 2*fibonacci(n-1)+fibonacci(n))
%o vector(50, n, lucas(n-1)^3-lucas(n-1)^2)
%o (Magma) [Lucas(n)^3 - Lucas(n)^2: n in [0..30]]; // _Vincenzo Librandi_, Jun 26 2014
%Y Cf. A000032, A045991, A244309.
%K nonn,easy
%O 0,1
%A _Colin Barker_, Jun 25 2014
|
