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A292696 a(n) = L(n)^2 - 5*(-1)^n = L(n+1)*L(n-1), where L = A000032.

%I A292696 #46 Feb 16 2025 08:33:51
%S A292696 -1,6,4,21,44,126,319,846,2204,5781,15124,39606,103679,271446,710644,
%T A292696 1860501,4870844,12752046,33385279,87403806,228826124,599074581,
%U A292696 1568397604,4106118246,10749957119,28143753126,73681302244,192900153621,505019158604,1322157322206,3461452807999
%N A292696 a(n) = L(n)^2 - 5*(-1)^n = L(n+1)*L(n-1), where L = A000032.
%C A292696 This is the case k=1 of the identity L(n)^2 + 5*F(k)^2*(-1)^(n+k) = L(n+k)*L(n-k), where F = A000045. See also the comment in A292612.
%D A292696 Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications (2008), page 29 (the formula 20b, for h=-k, gives the identity in Comments section).
%H A292696 Colin Barker, <a href="/A292696/b292696.txt">Table of n, a(n) for n = 0..1000</a>
%H A292696 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormulae.html#section5.3">Fibonacci and Lucas Numbers</a> (see 31st formula: the identity in the Comments section is the case i=-k).
%H A292696 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a> (see the formula n. 5).
%H A292696 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A292696 G.f.: (-1 + 8*x - 6*x^2)/((1 + x)*(1 - 3*x + x^2)).
%F A292696 a(n) = a(-n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A292696 a(n) = -A001654(n+1) + 8*A001654(n) - 6*A001654(n-1) with A001654(-1)=0.
%F A292696 a(n) = L(2*n) - 3*(-1)^n.
%F A292696 Sum_{i=0..n} a(i) = L(2*n+1) - (3*(-1)^n + 1)/2.
%F A292696 a(n) = 2^(-n)*(-3*(-2)^n + (3-sqrt(5))^n + (3+sqrt(5))^n). - _Colin Barker_, Sep 21 2017
%F A292696 From _Peter Bala_, Oct 14 2019: (Start)
%F A292696 Sum_{n >= 1} L(n)/a(n) = 3/2.
%F A292696 Sum_{n >= 1} (-1)^n*L(n)/a(n) = 1/2.  (End)
%F A292696 From _Amiram Eldar_, Oct 06 2020: (Start)
%F A292696 Sum_{n>=1} 1/a(n) = 1/2.
%F A292696 Sum_{n>=1} (-1)^n/a(n) = 7/10 - 2*phi/5, where phi is the golden ratio (A001622). (End)
%p A292696 A292696:=proc(n) option remember:
%p A292696 if n=0 then -1 elif n=1 then 6 elif n=2 then 4 elif  n>=3 then 2*procname(n-1)+2*procname(n-2)-procname(n-3) fi; end:
%p A292696 seq(A292696(n),n=0..10^2); # _Muniru A Asiru_, Oct 03 2017
%t A292696 Table[LucasL[n]^2 - 5 (-1)^n, {n, 0, 40}]
%t A292696 LinearRecurrence[{2,2,-1},{-1,6,4},40] (* _Harvey P. Dale_, Oct 02 2018 *)
%o A292696 (Sage) [lucas(n)^2-5*(-1)^n for n in range(40)]
%o A292696 (Magma) [Lucas(n)^2-5*(-1)^n: n in [0..40]];
%o A292696 (PARI) Vec(-(1 - 8*x + 6*x^2) / ((1 + x)*(1 - 3*x + x^2)) + O(x^30)) \\ _Colin Barker_, Sep 21 2017
%o A292696 (GAP)
%o A292696 a:=[-1,6,4];; for n in [4..10^3] do a[n]:= 2*a[n-1]+2*a[n-2]-a[n-3]; od; A292696:=a; # _Muniru A Asiru_, Oct 03 2017
%Y A292696 Cf. A000032, A001622, A001654, A292612.
%Y A292696 Cf. A059929: Fibonacci(n+2)*Fibonacci(n).
%K A292696 sign,easy
%O A292696 0,2
%A A292696 _Bruno Berselli_, Sep 21 2017