A163194 a(n) = F(n)^2 * L(n+1)^2 * F(n-1) * L(n+2), where F(n) and L(n) are the Fibonacci and Lucas numbers, respectively.

0, 0, 112, 2156, 39204, 704700, 12648640, 226979168, 4072998384, 73087049196, 1311494037700, 23533806023420, 422297015415552, 7577812474157376, 135978327526488304, 2440032083021144300, 43784599166902574820, 785682752921352087228, 14098504953417767184064, 252987406408599326907296

Offset

0,3

Comments

Natural bilateral extension (brackets mark index 0): ..., 39200, 2160, 108, 4, -4, [0], 0, 112, 2156, 39204, 704700, ... This is A163196-reversed followed by A163194. That is, A163194(-n) = A163196(n-1).

References

Stuart Clary and Paul D. Hemenway, On sums of cubes of Fibonacci numbers, Applications of Fibonacci Numbers, Vol. 5 (St. Andrews, 1992), 123-136, Kluwer Acad. Publ., 1993. See equation (3).

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (20,-35,-35,20,-1).

Formula

Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).

a(n) = F(n)^2 * L(n+1)^2 * F(n-1) * L(n+2).

Unfactored form: a(n) = (1/5)*(F(6n+3) - 12*F(2n+1) + 10*(-1)^n).

Unfactored form: a(n) = F(2n+1)^3 - 3*F(2n+1) + 2*(-1)^n.

Summation form: a(n) = 4*Sum_{k=1..n} F(2k)^3 = 4 A163198(n) if n is even; a(n) = 4*Sum_{k=2..n} F(2k)^3 = 4 A163199(n) if n is odd. a(n) - 21 a(n-1) + 56 a(n-2) - 21 a(n-3) + a(n-4) = 200*(-1)^n.

a(n) - 20 a(n-1) + 35 a(n-2) + 35 a(n-3) - 20 a(n-4) + a(n-5) = 0.

G.f.: (112*x^2 - 84*x^3 + 4*x^4)/(1 - 20*x + 35*x^2 + 35*x^3 - 20*x^4 + x^5) = 4*x^2*(28 - 21*x + x^2)/((1 + x)*(1 - 3*x + x^2)*(1 - 18*x + x^2)).

a(n) - A163196(n) = 4*(-1)^n.

Example

G.f. = 112*x^2 + 2156*x^3 + 39204*x^4 + 704700*x^5 + 12648640*x^6 + ...

Mathematica

a[n_Integer] := Fibonacci[n]^2 LucasL[n+1]^2 Fibonacci[n-1] LucasL[n+2]
LinearRecurrence[{20, -35, -35, 20, -1}, {0, 0, 112, 2156, 39204}, 50] (* or *) Table[(1/5)*(Fibonacci[6n+3] - 12*Fibonacci[2n+1] + 10*(-1)^n), {n, 0, 25}] (* G. C. Greubel, Dec 09 2016 *)

Prog

(PARI) for(n=0, 30, print1((1/5)*(fibonacci(6*n+3) - 12*fibonacci(2*n+1) + 10*(-1)^n), ", ")) \\ G. C. Greubel, Dec 21 2017
(Magma) [(Fibonacci(n)*Lucas(n+1))^2*(Fibonacci(n-1)*Lucas(n+2)): n in [0..30]]; // G. C. Greubel, Dec 21 2017

Crossrefs

Cf. A163195, A163196, A163197, A163198, A163199.

Sequence in context: A233884 A246895 A188153 * A267327 A008361 A271671

Adjacent sequences: A163191 A163192 A163193 * A163195 A163196 A163197

Keyword

nonn,easy

Author

Stuart Clary, Jul 24 2009

Status

approved