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A127595
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a(n) = F(4n) - 2F(2n) where F(n) = Fibonacci numbers A000045.
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5
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0, 1, 15, 128, 945, 6655, 46080, 317057, 2176335, 14925184, 102320625, 701373311, 4807434240, 32951037313, 225850798095, 1548007091840, 10610205501105, 72723448842367, 498453982018560, 3416454544730369, 23416728143799375
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OFFSET
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0,3
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COMMENTS
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a(n) is a divisibility sequence; that is, if h|k then a(h)|a(k).
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LINKS
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FORMULA
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a(2n) = 5*(F(2n))^3*L(2n), a(2n+1) = F(2n+1)*L(2n+1)^3.
a(n) = [(Phi^(2n))-1]^2*[(Phi^(4n))-1]/[sqrt(5)*(Phi^(4n))].
G.f.: A(x)=x*(1+(r+2)*x+x^2)/((1-r*x+x^2)*(1-(r^2-2)*x+x^2)) at r=3. The case r=2 is A000578.
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EXAMPLE
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G.f. = x + 15*x^2 + 128*x^3 + 945*x^4 + 6655*x^5 + ... - Michael Somos, Dec 30 2022
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MATHEMATICA
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With[{r = 3}, CoefficientList[Series[x (1 + (r + 2) x + x^2)/((1 - r x + x^2)*(1 - (r^2 - 2)*x + x^2)), {x, 0, 20}], x]] (* Michael De Vlieger, Nov 09 2021 *)
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PROG
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(PARI) {a(n) = my(w = quadgen(5)^(2*n)); imag(w^2 - 2*w)}; /* Michael Somos, Dec 30 2022 */
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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