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A288913
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a(n) = Lucas(4*n + 3).
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5
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4, 29, 199, 1364, 9349, 64079, 439204, 3010349, 20633239, 141422324, 969323029, 6643838879, 45537549124, 312119004989, 2139295485799, 14662949395604, 100501350283429, 688846502588399, 4721424167835364, 32361122672259149, 221806434537978679, 1520283919093591604
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OFFSET
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0,1
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COMMENTS
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LINKS
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FORMULA
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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+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.
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MATHEMATICA
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LucasL[4 Range[0, 21] + 3]
LinearRecurrence[{7, -1}, {4, 29}, 30] (* G. C. Greubel, Dec 22 2017 *)
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PROG
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(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
(Magma) [Lucas(4*n + 3): n in [0..30]]; // G. C. Greubel, Dec 22 2017
(Python)
from sympy import lucas
def a(n): return lucas(4*n + 3)
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CROSSREFS
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Partial sums are in A081007 (after 0).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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