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

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

Author

Lekraj Beedassy, May 14 2005

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000032, A000045, A000204, A001254, A081071, A104457.

Programs

Magma

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

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 *)

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

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)

From Enrique Navarrete, Mar 24 2025: (Start)

a(n)^2 = 20 + 5*A081071(n).

Limit_{n->oo} a(n+1)/a(n) = (3 + sqrt(5))/2 (see A104457). (End)

Extensions

Corrected by T. D. Noe, Dec 11 2006

More terms from Bruno Berselli, Jul 17 2011