%I #29 Jan 05 2025 19:51:41
%S 0,1,20,78,300,980,3120,9527,28560,84114,244750,704836,2013120,
%T 5709613,16097900,45155190,126099120,350765012,972333360,2687024099,
%U 7404969000,20355783546,55829853970,152808294148,417451104000,1138438728025,3099690135620,8427213134622
%N a(n) = n*(Lucas(n)*Lucas(n+1) - 2).
%H Robert Israel, <a href="/A292360/b292360.txt">Table of n, a(n) for n = 0..2381</a>
%H Florian Luca (editor), <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/FebAdvProbSoln2016.pdf">Problem H-755</a>, The Fibonacci Quarterly, Volume 54, Number 1 (February 2016), p. 92 (see solution by Adnan Ali).
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,-6,20,-6,-9,6,-1).
%F a(n) = n*Sum_{k=1..n} Lucas(k)^2 (by Adnan Ali, see References).
%F From _Colin Barker_, Sep 15 2017: (Start)
%F G.f.: x*(1 + 14*x - 33*x^2 + 18*x^3 - 18*x^4 + 14*x^5 - 4*x^6)/((1 - x)^2*(1 + x)^2*(1 - 3*x + x^2)^2).
%F a(n) = 6*a(n-1) - 9*a(n-2) - 6*a(n-3) + 20*a(n-4) - 6*a(n-5) - 9*a(n-6) + 6*a(n-7) - a(n-8) for n > 7. (End)
%F a(n) = n*(Lucas(2*n+1)-1) for n even, otherwise a(n) = n*(Lucas(2*n+1)-3). - _Bruno Berselli_, Sep 15 2017
%p lucas:= gfun:-rectoproc({a(n)=a(n-1)+a(n-2),a(0)=2,a(1)=1},a(n),remember):
%p f:= n -> n*(lucas(n)*lucas(n+1)-2):
%p map(f, [$0..100]); # _Robert Israel_, Sep 17 2017
%t Table[n (LucasL[n] LucasL[n+1] - 2), {n, 0, 40}]
%o (Magma) [n*(Lucas(n)*Lucas(n+1)-2): n in [0..35]];
%o (PARI) concat(0, Vec(x*(1 + 14*x - 33*x^2 + 18*x^3 - 18*x^4 + 14*x^5 - 4*x^6) / ((1 - x)^2*(1 + x)^2*(1 - 3*x + x^2)^2) + O(x^30))) \\ _Colin Barker_, Sep 15 2017
%o (PARI) a000032(n) = fibonacci(n+1)+fibonacci(n-1); a(n) = n*(a000032(n)*a000032(n+1)-2); \\ _Altug Alkan_, Sep 15 2017
%Y Cf. A000032, A002878.
%K nonn,easy
%O 0,3
%A _Vincenzo Librandi_, Sep 15 2017