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A081185
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8th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).
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12
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0, 1, 16, 194, 2112, 21764, 217280, 2127112, 20562432, 197117968, 1879016704, 17842953248, 168988216320, 1597548359744, 15083504344064, 142288071200896, 1341431869882368, 12641049503662336, 119088016125890560
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OFFSET
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0,3
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COMMENTS
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This sequence is part of a class of sequences defined by the recurrence a(n,m) = 2*(m+1)*a(n-1,m) - ((m+1)^2 - 2)*a(n-2,m) with a(0) = 1 and a(1) = m+9. The generating function is Sum_{n>=0} a(n,m)*x^n = (1 - (m-7)*x)/(1 - 2*(m+1)*x + ((m+1)^2 - 2)*x^2) and has a series expansion in terms of Pell-Lucas numbers defined by a(n, m) = (1/2)*Sum_{k=0..n} binomial(n,k)*m^(n-k)*(5*Q(k) + 4*Q(k-1)). - G. C. Greubel, Mar 12 2021
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LINKS
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FORMULA
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a(n) = 16*a(n-1) - 62*a(n-2), a(0)=0, a(1)=1.
G.f.: x/(1 - 16*x + 62*x^2).
a(n) = ((8 + sqrt(2))^n - (8 - sqrt(2))^n)/(2*sqrt(2)).
a(n) = Sum_{k=0..n} C(n,2*k+1) * 2^k * 7^(n-2*k-1).
a(n) = (1/2)*Sum_{k=0..n-1} binomial(n-1,k)*7^(n-k-1)*(5*Q(k) + 4*Q(k-1)), where Q(n) = Pell-Lucas(n) = A002203(n). (End)
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MAPLE
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m:=30; S:=series( x/(1-16*x+62*x^2), x, m+1):
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MATHEMATICA
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CoefficientList[Series[x/(1-16x+62x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{16, -62}, {0, 1}, 30] (* Harvey P. Dale, Sep 24 2013 *)
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PROG
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(Magma) [n le 2 select n-1 else 16*Self(n-1)-62*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 07 2013
(Sage) [( x/(1-16*x+62*x^2) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 12 2021
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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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