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A288913 a(n) = Lucas(4*n + 3). 4
4, 29, 199, 1364, 9349, 64079, 439204, 3010349, 20633239, 141422324, 969323029, 6643838879, 45537549124, 312119004989, 2139295485799, 14662949395604, 100501350283429, 688846502588399, 4721424167835364, 32361122672259149, 221806434537978679, 1520283919093591604 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n) mod 4 gives A101000.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (7,-1).

FORMULA

G.f.: (4 + x)/(1 - 7*x + x^2).

a(n) = 7*a(n-1) - a(n-2) for n>1, with a(0)=4, a(1)=29.

a(n) = ((sqrt(5) + 1)^(4*n + 3) - (sqrt(5) - 1)^(4*n + 3))/(8*16^n).

a(n) = Fibonacci(4*n+4) + Fibonacci(4*n+2).

a(n) = 4*A004187(n+1) + A004187(n).

a(n) = 5*A003482(n) + 4 = 5*A081016(n) - 1.

a(n) = A002878(2*n+1) = A093960(2*n+3) = A001350(4*n+3) = A068397(4*n+3).

a(n+1)*a(n+k) - a(n)*a(n+k+1) = 15*Fibonacci(4*k). Example: for k=6, a(n+1)*a(n+6) - a(n)*a(n+7) = 15*Fibonacci(24) = 695520.

MAPLE

with(combinat): seq(fibonacci(4*n+4)+fibonacci(4*n+2), n=0..21); # Paolo P. Lava, Jun 20 2017

MATHEMATICA

LucasL[4 Range[0, 21] + 3]

LinearRecurrence[{7, -1}, {4, 29}, 30] (* G. C. Greubel, Dec 22 2017 *)

PROG

(PARI) Vec((4 + x)/(1 - 7*x + x^2) + O(x^30)) \\ Colin Barker, Jun 20 2017

(Sage)

def L():

    x, y = -1, 4

    while true:

        yield y

        x, y = y, 7*y - x

r = L(); [r.next() for _ in (0..21)] # Peter Luschny, Jun 20 2017

(MAGMA) [Lucas(4*n + 3): n in [0..30]]; // G. C. Greubel, Dec 22 2017

CROSSREFS

Cf. A000032, A000045, A004187, A101000.

Cf. A033891: fourth quadrisection of A000045.

Partial sums are in A081007 (after 0).

Positive terms of A098149, and subsequence of A001350, A002878, A016897, A093960, A068397.

Quadrisection of A000032: A056854 (first), A056914 (second), A246453 (third, without 11), this sequence (fourth).

Sequence in context: A121191 A129587 A143551 * A100022 A001883 A281600

Adjacent sequences:  A288910 A288911 A288912 * A288914 A288915 A288916

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Jun 19 2017

STATUS

approved

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Last modified November 17 11:02 EST 2019. Contains 329226 sequences. (Running on oeis4.)