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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 (list; graph; refs; listen; history; text; internal format)
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
LINKS
Tanya Khovanova, Recursive Sequences
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
Sequence in context: A112024 A362002 A245415 * A212950 A038252 A211865
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

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Last modified March 29 09:14 EDT 2024. Contains 371268 sequences. (Running on oeis4.)