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A106729 Sum of two consecutive squares of Lucas numbers (A001254).

%I A106729 #80 Mar 31 2025 21:27:50
%S A106729 5,10,25,65,170,445,1165,3050,7985,20905,54730,143285,375125,982090,
%T A106729 2571145,6731345,17622890,46137325,120789085,316229930,827900705,
%U A106729 2167472185,5674515850,14856075365,38893710245,101825055370,266581455865
%N A106729 Sum of two consecutive squares of Lucas numbers (A001254).
%C A106729 Positive values of x (or y) satisfying x^2 - 3xy + y^2 + 25 = 0. - _Colin Barker_, Feb 08 2014
%C A106729 Positive values of x (or y) satisfying x^2 - 7xy + y^2 + 225 = 0. - _Colin Barker_, Feb 09 2014
%C A106729 Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 1600 = 0. - _Colin Barker_, Feb 26 2014
%H A106729 Bruno Berselli, <a href="/A106729/b106729.txt">Table of n, a(n) for n = 0..300</a>
%H A106729 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A106729 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).
%F A106729 a(n) = Lucas(n)^2 + Lucas(n+1)^2 = 5*(Fibonacci(n)^2 + Fibonacci(n+1)^2) = 5*A001519(n+1).
%F A106729 a(n) = 3*a(n-1) - a(n-2). - _T. D. Noe_, Dec 11 2006
%F A106729 G.f.: 5*(1-x)/(1-3*x+x^2). - _Philippe Deléham_, Nov 16 2008
%F A106729 a(n) = Fibonacci(n-2)^2 + Fibonacci(n+3)^2. - _Gary Detlefs_, Dec 28 2010
%F A106729 a(n) = [1,1; 1,2]^(n-2).{3,4}.{3,4}, for n>=3. - _John M. Campbell_, Jul 09 2011
%F A106729 a(n) = Lucas(2n) + Lucas(2n+2). - _Richard R. Forberg_, Nov 23 2014
%F A106729 From _Robert Israel_, Nov 23 2014: (Start)
%F A106729 a(n) = 5*A000045(2*n+1).
%F A106729 E.g.f.: (5+sqrt(5))/2 * exp((3+sqrt(5))*x/2) + (5-sqrt(5))/2 * exp((3-sqrt(5))*x/2). (End)
%F A106729 From _Enrique Navarrete_, Mar 24 2025: (Start)
%F A106729 a(n)^2 = 20 + 5*A081071(n).
%F A106729 Limit_{n->oo} a(n+1)/a(n) = (3 + sqrt(5))/2 (see A104457). (End)
%p A106729 seq(combinat:-fibonacci(n-2)^2 + combinat:-fibonacci(n+3)^2, n=0..100); # _Robert Israel_, Nov 23 2014
%t A106729 Table[LucasL[n]^2 + LucasL[n+1]^2, {n, 0, 30}] (* _Wesley Ivan Hurt_, Nov 23 2014 *)
%t A106729 Total/@Partition[LucasL[Range[0,30]]^2,2,1] (* _Harvey P. Dale_, Jun 26 2022 *)
%o A106729 (Magma) [Fibonacci(n-2)^2+Fibonacci(n+3)^2: n in [0..30]]; // _Vincenzo Librandi_, Jul 09 2011
%o A106729 (PARI) a(n) = fibonacci(n-2)^2 + fibonacci(n+3)^2;
%o A106729 vector(30, n, a(n-1)) \\ _G. C. Greubel_, Dec 17 2017
%o A106729 (Sage) [fibonacci(n-2)^2 + fibonacci(n+3)^2 for n in (0..30)] # _G. C. Greubel_, Sep 10 2021
%Y A106729 Cf. A000032, A000045, A000204, A001254, A081071, A104457.
%K A106729 nonn,easy
%O A106729 0,1
%A A106729 _Lekraj Beedassy_, May 14 2005
%E A106729 Corrected by _T. D. Noe_, Dec 11 2006
%E A106729 More terms from _Bruno Berselli_, Jul 17 2011