A292360 a(n) = n*(Lucas(n)*Lucas(n+1) - 2).

0, 1, 20, 78, 300, 980, 3120, 9527, 28560, 84114, 244750, 704836, 2013120, 5709613, 16097900, 45155190, 126099120, 350765012, 972333360, 2687024099, 7404969000, 20355783546, 55829853970, 152808294148, 417451104000, 1138438728025, 3099690135620, 8427213134622

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..2381

Florian Luca (editor), Problem H-755, The Fibonacci Quarterly, Volume 54, Number 1 (February 2016), p. 92 (see solution by Adnan Ali).

Index entries for linear recurrences with constant coefficients, signature (6,-9,-6,20,-6,-9,6,-1).

Formula

a(n) = n*Sum_{k=1..n} Lucas(k)^2 (by Adnan Ali, see References).

From Colin Barker, Sep 15 2017: (Start)

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).

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)

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

Maple

lucas:= gfun:-rectoproc({a(n)=a(n-1)+a(n-2), a(0)=2, a(1)=1}, a(n), remember):
f:= n -> n*(lucas(n)*lucas(n+1)-2):
map(f, [$0..100]); # Robert Israel, Sep 17 2017

Mathematica

Table[n (LucasL[n] LucasL[n+1] - 2), {n, 0, 40}]

Prog

(Magma) [n*(Lucas(n)*Lucas(n+1)-2): n in [0..35]];
(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
(PARI) a000032(n) = fibonacci(n+1)+fibonacci(n-1); a(n) = n*(a000032(n)*a000032(n+1)-2); \\ Altug Alkan, Sep 15 2017

Crossrefs

Cf. A000032, A002878.

Sequence in context: A266133 A083127 A211613 * A337988 A002609 A195322

Adjacent sequences: A292357 A292358 A292359 * A292361 A292362 A292363

Keyword

nonn,easy

Author

Vincenzo Librandi, Sep 15 2017

Status

approved