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A099157
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a(n) = 4^(n-1)*U(n-1, 3/2) where U is the Chebyshev polynomial of the second kind.
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4
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0, 1, 12, 128, 1344, 14080, 147456, 1544192, 16171008, 169345024, 1773404160, 18571329536, 194481487872, 2036636581888, 21327935176704, 223349036810240, 2338941478895616, 24493713157783552, 256501494231072768
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f.: x/(1-12*x+16*x^2).
E.g.f.: exp(6*x) * sinh(2*sqrt(5)*x)/sqrt(5).
a(n) = 12*a(n-1) - 16*a(n-2).
a(n) = sqrt(5)/20 * ( (sqrt(5)+1)^(2*n) - (sqrt(5)-1)^(2*n) ).
a(n) = Sum_{k=0..n} binomial(2*n, 2*k+1) * 5^k / 2.
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MATHEMATICA
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M= {{0, 1}, {-16, 12}}; v[0] = {0, 1}; v[n_]:= v[n]= M.v[n-1];
LinearRecurrence[{12, -16}, {0, 1}, 20] (* Harvey P. Dale, Sep 27 2015 *)
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PROG
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(Sage) [lucas_number1(n, 12, 16) for n in range(0, 19)] # Zerinvary Lajos, Apr 27 2009
(PARI) a(n) = 4^(n-1)*polchebyshev(n-1, 2, 3/2); \\ Michel Marcus, Jun 10 2018
(Magma) [n le 2 select n-1 else 12*Self(n-1) -16*Self(n-2): n in [1..30]]; // G. C. Greubel, Jul 20 2023
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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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EXTENSIONS
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STATUS
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approved
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