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A099922
a(n) = F(4n) - 2n, where F(n) = Fibonacci numbers A000045.
1
1, 17, 138, 979, 6755, 46356, 317797, 2178293, 14930334, 102334135, 701408711, 4807526952, 32951280073, 225851433689, 1548008755890, 10610209857691, 72723460248107, 498454011879228, 3416454622906669, 23416728348467645
OFFSET
1,2
REFERENCES
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 54.
FORMULA
G.f.: x*(1+8*x+x^2)/((1-x)^2 * (1-7*x+x^2)). [Corrected for offset by Georg Fischer, May 24 2019]
a(n) = Sum_{k=1..n} Lucas(2k-1)^2.
From Colin Barker, May 25 2019: (Start)
a(n) = (-((7-3*sqrt(5))/2)^n + ((7+3*sqrt(5))/2)^n)/sqrt(5) - 2*n.
a(n) = 9*a(n-1) - 16*a(n-2) + 9*a(n-3) - a(n-4) for n>4.
(End)
MATHEMATICA
Rest[CoefficientList[Series[x*(1+8x+x^2)/((1-x)^2*(1-7x+x^2)), {x, 0, 20}], x]] (* Georg Fischer, May 24 2019 *)
PROG
(PARI) Vec(x*(1 + 8*x + x^2) / ((1 - x)^2*(1 - 7*x + x^2)) + O(x^25)) \\ Colin Barker, May 25 2019
CROSSREFS
Equals A033888(n) - 2n. Partial sums of A081071. Bisection of A054452.
Sequence in context: A271395 A358202 A277387 * A298626 A157360 A142815
KEYWORD
nonn
AUTHOR
Ralf Stephan, Nov 01 2004
STATUS
approved