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A236144
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a(n) = F(floor( (n+3)/2 )) * L(floor( (n+2)/2 )) where F=Fibonacci and L=Lucas numbers.
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1
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2, 2, 1, 2, 6, 9, 12, 20, 35, 56, 88, 143, 234, 378, 609, 986, 1598, 2585, 4180, 6764, 10947, 17712, 28656, 46367, 75026, 121394, 196417, 317810, 514230, 832041, 1346268, 2178308, 3524579, 5702888, 9227464, 14930351, 24157818, 39088170, 63245985, 102334154
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OFFSET
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0,1
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LINKS
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FORMULA
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G.f.: (2 - x^2 - x^3) / (1 - x - x^3 - x^4) = (1 - x) * (2 + 2*x + x^2) / ((1 + x^2) * (1 - x - x^2)).
a(n) = a(n-1) + a(n-3) + a(n-4) for all n in Z.
0 = a(n)*a(n+2) + a(n+1)*(+a(n+2) -a(n+3)) for all n in Z.
a(2*n - 1) = F(n+1) * L(n-1) = A128535(n+1). a(2*n) = F(n+1) * L(n) = A128534(n+1).
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EXAMPLE
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G.f. = 2 + 2*x + x^2 + 2*x^3 + 6*x^4 + 9*x^5 + 12*x^6 + 20*x^7 + 35*x^8 + ...
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MATHEMATICA
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a[ n_] := Fibonacci[ Quotient[ n + 3, 2]] LucasL[ Quotient[ n, 2]];
CoefficientList[Series[(2-x^2-x^3)/(1-x-x^3-x^4), {x, 0, 60}], x] (* G. C. Greubel, Aug 07 2018 *)
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PROG
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(PARI) {a(n) = fibonacci( (n+3)\2 ) * (fibonacci( n\2+1 ) + fibonacci( n\2-1 ))};
(PARI) x='x+O('x^60); Vec((2-x^2-x^3)/(1-x-x^3-x^4) \\ G. C. Greubel, Aug 07 2018
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2-x^2-x^3)/(1-x-x^3-x^4)); // G. C. Greubel, Aug 07 2018
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CROSSREFS
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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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