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A106729
Sum of two consecutive squares of Lucas numbers (A001254).
9
5, 10, 25, 65, 170, 445, 1165, 3050, 7985, 20905, 54730, 143285, 375125, 982090, 2571145, 6731345, 17622890, 46137325, 120789085, 316229930, 827900705, 2167472185, 5674515850, 14856075365, 38893710245, 101825055370, 266581455865
OFFSET
0,1
COMMENTS
Positive values of x (or y) satisfying x^2 - 3xy + y^2 + 25 = 0. - Colin Barker, Feb 08 2014
Positive values of x (or y) satisfying x^2 - 7xy + y^2 + 225 = 0. - Colin Barker, Feb 09 2014
Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 1600 = 0. - Colin Barker, Feb 26 2014
FORMULA
a(n) = Lucas(n)^2 + Lucas(n+1)^2 = 5*(Fibonacci(n)^2 + Fibonacci(n+1)^2) = 5*A001519(n+1).
a(n) = 3*a(n-1) - a(n-2). - T. D. Noe, Dec 11 2006
G.f.: 5*(1-x)/(1-3*x+x^2). - Philippe Deléham, Nov 16 2008
a(n) = Fibonacci(n-2)^2 + Fibonacci(n+3)^2. - Gary Detlefs, Dec 28 2010
a(n) = [1,1; 1,2]^(n-2).{3,4}.{3,4}, for n>=3. - John M. Campbell, Jul 09 2011
a(n) = Lucas(2n) + Lucas(2n+2). - Richard R. Forberg, Nov 23 2014
From Robert Israel, Nov 23 2014: (Start)
a(n) = 5*A000045(2*n+1).
E.g.f.: (5+sqrt(5))/2 * exp((3+sqrt(5))*x/2) + (5-sqrt(5))/2 * exp((3-sqrt(5))*x/2). (End)
MAPLE
seq(combinat:-fibonacci(n-2)^2 + combinat:-fibonacci(n+3)^2, n=0..100); # Robert Israel, Nov 23 2014
MATHEMATICA
Table[LucasL[n]^2 + LucasL[n+1]^2, {n, 0, 30}] (* Wesley Ivan Hurt, Nov 23 2014 *)
Total/@Partition[LucasL[Range[0, 30]]^2, 2, 1] (* Harvey P. Dale, Jun 26 2022 *)
PROG
(Magma) [Fibonacci(n-2)^2+Fibonacci(n+3)^2: n in [0..30]]; // Vincenzo Librandi, Jul 09 2011
(PARI) a(n) = fibonacci(n-2)^2 + fibonacci(n+3)^2;
vector(30, n, a(n-1)) \\ G. C. Greubel, Dec 17 2017
(Sage) [fibonacci(n-2)^2 + fibonacci(n+3)^2 for n in (0..30)] # G. C. Greubel, Sep 10 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Lekraj Beedassy, May 14 2005
EXTENSIONS
Corrected by T. D. Noe, Dec 11 2006
More terms from Bruno Berselli, Jul 17 2011
STATUS
approved