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A014731

Squares of even Lucas numbers.

4, 16, 324, 5776, 103684, 1860496, 33385284, 599074576, 10749957124, 192900153616, 3461452808004, 62113250390416, 1114577054219524, 20000273725560976, 358890350005878084, 6440026026380244496, 115561578124838522884, 2073668380220713167376

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OFFSET

0,1

LINKS

Colin Barker, Table of n, a(n) for n = 0..700

R. S. Melham and A. G. Shannon, Inverse trigonometric and hyperbolic summation formulas involving generalized Fibonacci numbers, The Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.

Index entries for linear recurrences with constant coefficients, signature (17,17,-1).

FORMULA

a(n) = Fibonacci(6*n+3) - 2*Fibonacci(6*n) + 2*(-1)^n. - Ralf Stephan, May 14 2004

G.f.: 4*(-4*x^2-13*x+1)/((1+x)*(1-18*x+x^2)). - Ralf Stephan, May 14 2004

From Colin Barker, Mar 04 2016: (Start)

a(n) = 2*(-1)^n+(9+4*sqrt(5))^(-n)+(9+4*sqrt(5))^n.

a(n) = 17*a(n-1)+17*a(n-2)-a(n-3) for n>2. (End)

a(n) = A014448(n)^2. - Sean A. Irvine, Nov 18 2018

a(n) = 5*Fibonacci(3*n)^2 + 4*(-1)^n. - Amiram Eldar, Jan 11 2022

From Amiram Eldar, Jan 02 2026: (Start)

Sum_{n>=1} (-1)^(n+1) * arctanh(5/a(n)) = log(3*(phi-1))/2 (Melham and Shannon, 1995, p. 39, eq. (4.7)).

Product_{n>=1} (a(n)-(-1)^n*5)/(a(n)+(-1)^n*5) = 3*(phi-1) (A134973) (Melham and Shannon, 1995, p. 39, eq. (4.8)). (End)

MATHEMATICA

(Table[LucasL@ n, {n, 0, 52}] /. n_ /; OddQ@ n -> Nothing)^2 (* Michael De Vlieger, Mar 04 2016 *)

LinearRecurrence[{17, 17, -1}, {4, 16, 324}, 20] (* Harvey P. Dale, Nov 19 2024 *)

PROG

(PARI) Vec(4*(1-13*x-4*x^2)/((1+x)*(1-18*x+x^2)) + O(x^20)) \\ Colin Barker, Mar 04 2016

CROSSREFS

Cf. A000045, A001622 (phi), A014448, A134973.

Sequence in context: A203105 A273474 A095956 * A023114 A207851 A202681

Adjacent sequences: A014728 A014729 A014730 * A014732 A014733 A014734

KEYWORD

nonn,easy

AUTHOR

Mohammad K. Azarian

EXTENSIONS

More terms from Erich Friedman.

STATUS

approved

URL: https://oeis.org/A014731

⇱ A014731 - OEIS