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A106729 Sum of two consecutive squares of Lucas numbers (A001254). 8
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

Bruno Berselli, Table of n, a(n) for n = 0..300

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (3,-1).

FORMULA

a(n) = L(n)^2 + L(n+1)^2 = 5*(F(n)^2 + F(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) = (5/2)*((3/2)+(1/2)*sqrt(5))^n+(1/2)*((3/2)+(1/2)*sqrt(5))^n*sqrt(5)-(1/2)*((3/2)-(1/2)*sqrt(5))^n *sqrt(5)+(5/2)*((3/2)-(1/2)*sqrt(5))^n, with n>=0. - Paolo P. Lava, Nov 19 2008

a(n) = Fibonacci(n-2)^2 + Fibonacci(n+3)^2. - Gary Detlefs, Dec 28 2010

For n>=3, a(n)=[1,1;1,2]^(n-2).{3,4}.{3,4}. - John M. Campbell, Jul 09 2011

a(n) = L(2n) + L(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 *)

PROG

(MAGMA) [Fibonacci(n-2)^2+Fibonacci(n+3)^2: n in [0..30]]; // Vincenzo Librandi, Jul 09 2011

(PARI) for(n=0, 30, print1(fibonacci(n-2)^2 + fibonacci(n+3)^2, ", ")) \\ G. C. Greubel, Dec 17 2017

CROSSREFS

Cf. A000204.

Sequence in context: A025625 A112024 A245415 * A212950 A038252 A211865

Adjacent sequences:  A106726 A106727 A106728 * A106730 A106731 A106732

KEYWORD

nonn,easy,changed

AUTHOR

Lekraj Beedassy, May 14 2005

EXTENSIONS

Corrected by T. D. Noe, Dec 11 2006

More terms by Bruno Berselli, Jul 17 2011

STATUS

approved

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Last modified December 18 03:36 EST 2017. Contains 296128 sequences.