A099922 a(n) = F(4n) - 2n, where F(n) = Fibonacci numbers A000045.

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.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (9,-16,9,-1)

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

Adjacent sequences: A099919 A099920 A099921 * A099923 A099924 A099925

Keyword

nonn

Author

Ralf Stephan, Nov 01 2004

Status

approved