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A215042 a(n) = F(8*n)/L(2*n) with n >= 0, F = A000045 (Fibonacci numbers) and L = A000032 (Lucas numbers). 2
0, 7, 141, 2576, 46347, 831985, 14930208, 267913919, 4807525989, 86267568688, 1548008749155, 27777890017577, 498454011832896, 8944394323670071, 160500643816049277, 2880067194369984080, 51680708854856144763, 927372692193073296289 (list; graph; refs; listen; history; text; internal format)
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
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)).
Sequence in context: A306628 A302912 A191956 * A221267 A070074 A051397
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 31 2012
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)