A215042 a(n) = F(8*n)/L(2*n) with n >= 0, F = A000045 (Fibonacci numbers) and L = A000032 (Lucas numbers).

0, 7, 141, 2576, 46347, 831985, 14930208, 267913919, 4807525989, 86267568688, 1548008749155, 27777890017577, 498454011832896, 8944394323670071, 160500643816049277, 2880067194369984080, 51680708854856144763, 927372692193073296289

Offset

0,2

Comments

This provides the second example for the Riordan transition matrix R mentioned in a comment to A078812 (here the column called there n=1 is relevant).

Links

Table of n, a(n) for n=0..17.

Index entries for linear recurrences with constant coefficients, signature (21,-56,21,-1).

Formula

a(n) = 2*F(2*n) + 1*5*F(2*n)^3, n >= 0 (for the coefficients 2, 1, see the second row of the Riordan matrix R = A078812 (with offset [0,0])).

a(n) = F(6*n) - F(2*n), n >= 0, (from the preceding line and a 5*F(2*n)^3 formula given in a comment on the signed triangle A111418, with l->2*n, n->1; see also 5*A215039).

O.g.f.: x*(7-6*x+7*x^2)/((1-3*x+x^2)*(1-18*x+x^2)). The partial fraction decomposition and recurrences lead to the preceding formula.

Mathematica

Table[Fibonacci[8 n]/LucasL[2 n], {n, 0, 17}] (* Bruno Berselli, Aug 31 2012 *)
LinearRecurrence[{21, -56, 21, -1}, {0, 7, 141, 2576}, 20] (* Harvey P. Dale, Jul 18 2021 *)

Prog

(Magma) [Fibonacci(8*n)/Lucas(2*n): n in [0..17]]; // Bruno Berselli, Aug 31 2012

Crossrefs

Cf. A001906 (for F(4*n)/L(2*n) = F(2*n)), 24*A215043 (for F(12*n)/L(2*n)).

Cf. A000032, A000045.

Sequence in context: A306628 A302912 A191956 * A221267 A070074 A051397

Adjacent sequences: A215039 A215040 A215041 * A215043 A215044 A215045

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2012

Status

approved