A153173 a(n) = L(5*n)/L(n) where L(n) = Lucas number A000204(n).

11, 41, 341, 2161, 15251, 103361, 711491, 4868641, 33391061, 228811001, 1568437211, 10749853441, 73681573691, 505018447961, 3461454668501, 23725145626561, 162614613425891, 1114577020834241, 7639424866266611

Offset

1,1

Comments

All numbers in this sequence are congruent to 1 mod 10.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

L. Carlitz, Problem B-185, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 8, No. 3 (1970), p. 325; Lucas Ratio I, Solution to Problem B-185 by C. B. A. Peck, ibid., Vol. 9, No. 1 (1971), p. 109.

L. Carlitz, Problem B-186, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 8, No. 3 (1970), p. 326; Lucas Ratio II, Solution to Problem B-186 by John Wessner, ibid., Vol. 9, No. 1 (1971), pp. 109-110.

Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Formula

From R. J. Mathar, Oct 22 2010: (Start)

a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5).

G.f.: -x*(11-14*x-29*x^2+6*x^3+x^4)/( (x-1)*(x^2-7*x+1)*(x^2+3*x+1) ).

a(n) = 1 + A056854(n) - (-1)^n*A005248(n). (End)

From Amiram Eldar, Feb 02 2022: (Start)

a(n) = Lucas(2*n)^2 - (-1)^n*Lucas(2*n) - 1 (Carlitz, Problem B-185).

a(n) = (Lucas(2*n) - 3*(-1)^n)^2 + (-1)^n*(5*Fibonacci(n))^2 (Carlitz, Problem B-186). (End)

a(n) = a(-n) = 1 + 10*A085695(n) = 5 + L(n-1)*L(n)^2*L(n+1) for all n in Z. - Michael Somos, Apr 23 2022

Mathematica

Table[LucasL[5*n]/LucasL[n], {n, 1, 50}]
CoefficientList[Series[x*(11-14*x-29*x^2+6*x^3+x^4)/((1-x)*(x^2-7*x+1)*(x^2+3*x+1)), {x, 0, 50}], x] (* G. C. Greubel, Dec 21 2017 *)
a[ n_] := 1 + 5*Fibonacci[n]*Fibonacci[3*n]; (* Michael Somos, Apr 23 2022 *)

Prog

(PARI) {L(n)=fibonacci(n-1)+fibonacci(n+1)}; a(n) = L(5*n)/L(n) \\ Charles R Greathouse IV, Jun 11 2015
(PARI) my(x='x+O('x^30)); Vec(x*(11-14*x-29*x^2+6*x^3+x^4 )/((1-x)*(x^2-7*x +1)*(x^2+3*x+1))) \\ G. C. Greubel, Dec 21 2017
(PARI) {a(n) = 1 + 5*fibonacci(n)*fibonacci(3*n)}; /* Michael Somos, Apr 23 2022 */
(Magma) I:=[11, 41, 341, 2161, 15251]; [n le 5 select I[n] else 5*Self(n-1)+15*Self(n-2)-15*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 21 2017

Crossrefs

Cf. A000032, A000045, A000204, A005248, A056854, A085695, A110391.

Sequence in context: A092445 A068840 A082424 * A050526 A257967 A104118

Adjacent sequences: A153170 A153171 A153172 * A153174 A153175 A153176

Keyword

nonn,easy

Author

Artur Jasinski, Dec 20 2008

Status

approved