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A088139
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a(n) = 2*a(n-1) - 6*a(n-2), a(0)=0, a(1)=1.
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8
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0, 1, 2, -2, -16, -20, 56, 232, 128, -1136, -3040, 736, 19712, 35008, -48256, -306560, -323584, 1192192, 4325888, 1498624, -22958080, -54907904, 27932672, 385312768, 603029504, -1105817600, -5829812224, -5024718848, 24929435648, 80007184384
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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/(1-2*x+6*x^2).
E.g.f.: exp(x)*sin(sqrt(5)*x)/sqrt(5).
a(n) = ((1+i*sqrt(5))^n-(1-i*sqrt(5))^n)/(2*i*sqrt(5)).
a(n) = Im{(1+i*sqrt(5))^n/sqrt(5)}.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k+1)(-5)^k.
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MAPLE
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seq(coeff(series(x/(1-2*x+6*x^2), x, n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 23 2018
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MATHEMATICA
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TrigExpand@Table[(6^(n/2) Sin[n ArcTan[Sqrt[5]]])/Sqrt[5], {n, 0, 20}] (* or *)
Table[Sum[(-5)^k Binomial[n, 2 k + 1], {k, 0, n/2}], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 20 2016 *)
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PROG
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(Sage) [lucas_number1(n, 2, 6) for n in range(0, 30)] # Zerinvary Lajos, Apr 23 2009
(PARI) x='x+O('x^30); concat([0], Vec(x/(1-2*x+6*x^2))) \\ G. C. Greubel, Oct 22 2018
(Magma) I:=[0, 1]; [n le 2 select I[n] else 2*Self(n-1) - 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Oct 22 2018
(GAP) a:=[0, 1];; for n in [3..30] do a[n]:=2*a[n-1]-6*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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